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MK · lesson 10 of 14 · 43 min

Triangles

Triangles appear in a wide range of geometry questions, and this lesson gathers the essential facts in one place. It covers triangle anatomy and labeling, the interior angle-sum theorem, angles written as algebraic expressions, classification by sides and by angles, isosceles base angles, the exterior angle theorem, and how angle size relates to the opposite side.

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Video lesson: Triangles

Video transcript

0:00 Welcome to this explainer. Today we're taking on high school geometry, specifically triangles, and we're going to do it completely bare-handed. That's right, absolutely no calculators allowed. Whether you're studying for an exam that strictly restricts device usage, or maybe you just want to dramatically speed up your math reasoning, we'll equip you with a mental toolkit to instantly decode literally any triangle geometry problem. By the time we wrap up, you're going to look at these shapes in a completely different way. So let's start right in. Here's the blueprint for our toolkit. We'll start with anatomy and angle basics, then move to boundaries and space, open up the right triangle playbook, cover congruence and similarity rules, and finally look at high-value test traps. Let's kick things off with section 1, anatomy and angle basics. Okay, 180. This is hands-down the most important number in all of triangle geometry. The three interior angles of any triangle, no matter how stretched, squished or skewed it is, will always add up to exactly 180 degrees. If you have two angles, measuring, say, 48 and 67 degrees, you instantly know the third has to be 65.

1:00 If you want a really cool way to visualize why this works, imagine tearing the three corners off a paper triangle and lining them up side by side on a desk. Together, their edges will always form a perfectly straight line, and this brilliantly illustrates why when tests try to throw algebra at you, saying the angles are x, 2x and x plus 20, all you got to do is add those expressions up, set them equal to 180 and solve for x. Easy, right? Now, when you're classifying shapes, spotting an isosceles triangle is a massive shortcut. It's basically a cheat code. Because it has at least two equal sides, the angles directly opposite those sides, we call them the base angles, are exactly the same. Why does this matter to us? Well, because finding just one base angle instantly gives you the other. If a test tells you the top vertex angle is 50 degrees, you just subtract that 50 from 180 to get 130, cut it in half, and boom, your two base angles are 65 degrees each, just one tiny clue unlocks the whole shape. Boundaries and space.

2:04 Imagine you're handed three side lengths, let's say 5, 7 and 13, and you're asked if they can actually connect to form a valid closed triangle. What do you think? The answer is no way. They completely fail the strict triangle inequality rule. This rule is absolute. The two shorter sides of a triangle must add up to strictly more than the longest side. Think about it. If you take sticks of length 5 and 7 and try to connect them across a 13 length gap, they'll just lay flat. I mean, 5 plus 7 is only 12. They won't even touch, let alone peak to form an angle. So whenever you're testing side lengths, always add the two smaller guys. If they don't beat the big guy, it is definitely not a triangle. So once you've confirmed you actually have a valid triangle, you have to measure it. Perimeter is super simple. Just add up the outer side lengths like you're walking along a fence line. But area, that's where people get tripped up. The formula is one half the base times the perpendicular height. The crucial word there is perpendicular. Never ever let a slanted side trick you into using it as your height measurement.

3:08 The true height is the straight vertical drop from the top vertex down to the base, hitting it at a perfect 90 degree right angle. And why the one half? Well, any triangle is exactly half of a parallelogram with the very same base and height. Keep that specific visual in your mind and you'll literally never forget the fraction. Alright, let's get into the heavy hitter. Section 3, the right triangle playbook. A squared plus B squared equals C squared. Now, that might sound intimidating as an algebraic equation. But in practice, without a calculator, it's just a few simple mental steps. Let's walk through it. To find the longest side, the hypotenuse, you square the two shorter legs, add them together and take the square root. But what if you already have the hypotenuse and need a missing leg? You just reverse it. Square the hypotenuse, subtract the square of the known leg and take the square root. For example, say you're leaning a 26-foot ladder against a wall with the base 10 feet out. Square 26 to get 676, square 10 to get 100, subtract that 100 to get 576. The square root of 576 is 24.

4:11 Bam, the ladder reaches exactly 24 feet high. But let's be real. Doing all that squaring and square rooting by hand takes up precious time. That is exactly why memorizing Pythagorean triples, like 3, 4, 5, 5, 12, 13, and 8, 15, 17, is the ultimate time-saving hack. It literally lets you skip the square roots entirely. Test makers absolutely love these clean whole numbers. If you see a right triangle with legs of 3 and 4, you instantly know the hypotenuse is 5. And it gets better because any scaled multiple works too. If you see legs of 30 and 40, you don't need to square those massive numbers. You just recognize it's a 3, 4, 5 triangle multiplied by 10, meaning your hypotenuse is exactly 50. And the shortcuts just keep coming. You've got to keep an eye out for these two special right triangles because their predictable side ratios really allow you to unlock all missing sides from just a single clue. First, for a 45, 45, 90 isosceles-right triangle, the legs are exactly the same.

5:14 And the hypotenuse is just a leg multiplied by the square root of 2. Okay, take a breath because the second one is a little different. For a 30, 60, 90 triangle, the short leg is exactly half of the hypotenuse. And the long leg, it's just the short leg multiplied by the square root of 3. So if they tell you the hypotenuse is 18, you immediately know the short leg is 9 and the long leg is 9 times root 3, completely bypassing the Pythagorean theorem. Next up, section 4, congruence and similarity rules. The terminology here is absolutely vital. Think of congruent triangles as identical twins. They have the exact same shape and the exact same size. All the angles match, all the side lengths match. Now, similar triangles, on the other hand, are like clones that have been hit with a shrink ray or an enlarging beam. They maintain the exact same shape, meaning all their interior angles are completely identical, but they are scaled to different sizes. So while congruent triangles are perfectly equal, similar triangles use a scale factor, which we usually call K, to determine exactly how much larger or smaller their sides are compared to the original.

6:19 But listen up, because here is exactly where test makers try to trap you. When you scale a similar triangle by a factor of K, every length scales by that exact number. The sides, the perimeter, the height, all of it. But area? Area is a two-dimensional measurement of base times height, so it scales by K squared. This is absolutely crucial. If you triple the side lengths of a triangle, meaning K equals 3, the area doesn't just triple. It scales by 3 squared. It becomes 9 times larger. If you have a small triangle with an area of 20 square meters and you triple its sides, the new area isn't 60. It's 20 times 9, which is 180 square meters. Don't let them catch you slipping on that one. Finally, we arrive at section 5, high-value test traps. Treat these points as your ultimate minefield survival checklist. First up, trusting visual diagrams. Never, ever assume a triangle as a right triangle just because it kind of looks like one. Always base your conclusions on hard numbers or that little square marking in the corner. Second trap, using a slanted side for height when you're finding area.

7:25 Remember, height must always be perfectly perpendicular to the base. Third, forgetting the one-half in the area formula. Doing that will just give you the area of a rectangle and completely double your intended answer. Fourth, when using the Pythagorean theorem, just adding the legs together instead of squaring them first. And lastly, thinking side-side angle or angle angle angle prove congruence. Side-side angle can actually fit two totally different triangles. And angle-angle-angle only guarantees they are similar in shape, not that they are the exact same size. So with all these empowering tools locked into your mind, you can confidently solve basically any geometry challenge without ever reaching for a calculator. You understand the boundaries. The unbreakable 180-degree rule, the magical right triangle, cheat codes, and the sneaky traps they try to set. So the very next time you see a complicated messy shape on a test page, ask yourself, what is this triangle actually trying to hide from me? And which mental tool am I going to use to reveal it?

Contents · 41 sections
  1. Learning Outcomes, Level, and Honest Exam Context
  2. Triangle Anatomy and Labeling
  3. Interior Angle-Sum Theorem
  4. Angles Defined by Algebraic Expressions
  5. Side-Based Classification
  6. Angle-Based Classification
  7. Isosceles Base Angles
  8. Exterior Angle Theorem
  9. Angle Size and Opposite Side Length
  10. Triangle Inequality: Can These Sides Form a Triangle?
  11. Integer Side Constraints
  12. Perimeter of a Triangle
  13. Area: Half Base Times Perpendicular Height
  14. Area of Right and Obtuse Triangles
  15. Why the Area Formula Works
  16. Right Triangle Vocabulary
  17. Pythagorean Theorem: Find Hypotenuse
  18. Pythagorean Theorem: Find a Missing Leg
  19. Converse of the Pythagorean Theorem
  20. Recognize Useful Pythagorean Triples
  21. Congruent Triangles
  22. Similar Triangles
  23. Similarity Criteria
  24. Similarity and Perimeter Ratios
  25. Similarity and Area Ratios
  26. Special Right Triangle: 45°–45°–90°
  27. Special Right Triangle: 30°–60°–90°
  28. Angles From a Diagram: Facts Versus Appearance
  29. Combined Triangle Angle and Length Problems
  30. No-Calculator Triangle Toolkit
  31. Worked Example 1 — Unknown Algebraic Angles
  32. Worked Example 2 — Exterior Angle and Adjacent Interior
  33. Worked Example 3 — Strict Side-Length Bounds
  34. Worked Example 4 — Isosceles Angles
  35. Worked Example 5 — A Right Triangle With Missing Leg
  36. Worked Example 6 — Special Right Triangles
  37. Worked Example 7 — Similar Triangles
  38. Worked Example 8 — Area and Perpendicular Height
  39. Worked Example 9 — Similarity Changes Area
  40. High-Value Triangle Traps
  41. Triangle Formula and Reasoning Review
Visual lesson

Lesson slides

15 slides
Triangles – slide 1 of 15

Learning Outcomes, Level, and Honest Exam Context

This lesson covers triangle skills that can appear in ASVAB Mathematics Knowledge: the 180-degree angle sum, exterior angles, isosceles base angles, classification, the triangle inequality, perimeter and area, the Pythagorean theorem and its converse, congruence and similarity, and the 45-45-90 and 30-60-90 special right triangles. The level is high-school geometry. The test samples a broad range of topics, so expect some of these ideas rather than all of them on any one form.

Triangle Anatomy and Labeling

A triangle has three vertices, three sides, and three interior angles. Vertices are named with capital letters, as in triangle ABC. The side opposite a vertex is often labeled with the matching lowercase letter, so side a lies across from angle A. Each side joins two vertices: side BC connects B and C and is the side opposite angle A.

Triangle ABC with sides labeled a, b and c; side a (BC) and angle A are highlighted to show that side a lies opposite angle A.

Interior Angle-Sum Theorem

The three interior angles of any triangle add up to 180 degrees. If two angles are 48 and 67 degrees, the third is

180 - 48 - 67 = 65 degrees

One way to see why: tear the three corners off a paper triangle and line them up; together they form a straight line.

A triangle with angles 48°, 67° and 65°, and the same three angles placed side by side to fill a straight line: 48 + 65 + 67 = 180°.

Angles Defined by Algebraic Expressions

When angles are written as expressions, add them and set the sum equal to 180. For angles of x, 2x and (x + 20) degrees,

4x + 20 = 180

so x = 40. The angles are 40, 80 and 60 degrees. Always substitute back to find each angle and confirm the total is 180.

Side-Based Classification

By sides, a triangle is equilateral if all three sides are equal, isosceles if at least two sides are equal, and scalene if no two sides are equal. A triangle with sides 6, 6 and 10 is isosceles; 7, 8 and 11 is scalene; 5, 5 and 5 is equilateral, which also makes each of its angles 60 degrees.

Three triangles drawn to scale: isosceles with sides 6, 6 and 10, scalene with sides 7, 8 and 11, and equilateral with sides 5, 5 and 5; tick marks show the equal sides.

Angle-Based Classification

By angles, a triangle is acute if all three angles are less than 90 degrees, right if one angle is exactly 90 degrees, and obtuse if one angle is greater than 90 degrees. A triangle can have at most one right or obtuse angle, because two such angles alone would reach at least 180 degrees.

Three triangles: acute with all angles under 90°, right with one 90° angle marked by a square, and obtuse with one angle over 90° marked.

Isosceles Base Angles

In an isosceles triangle, the angles opposite the two equal sides, called base angles, are equal. The third angle is the vertex angle. If the vertex angle is 50 degrees, each base angle is

(180 - 50)/2 = 65 degrees

If a base angle is 72 degrees, the vertex angle is

180 - 144 = 36 degrees
Two isosceles triangles with tick marks on the equal sides: one with vertex angle 50° and base angles 65° and 65°, the other with base angles 72° and 72° and vertex angle 36°.

Exterior Angle Theorem

Extend one side of a triangle past a vertex, and the angle formed outside is an exterior angle. It forms a linear pair with the interior angle beside it, so those two add to 180 degrees. It also equals the sum of the two remote interior angles, the ones not touching it. If the remote angles are 38 and 77 degrees, the exterior angle is 115 degrees and the adjacent interior angle is 65 degrees.

Triangle with interior angles 38°, 77° and 65°; one side is extended to form an exterior angle of 115°, which equals 38 + 77 and forms a straight line with the 65° angle.

Angle Size and Opposite Side Length

In any triangle, the largest angle lies opposite the longest side and the smallest angle lies opposite the shortest side. If triangle DEF has angles D = 40, E = 65 and F = 75 degrees, the longest side is DE, opposite F, and the shortest is EF, opposite D. In a right triangle, the hypotenuse is always the longest side.

Triangle DEF drawn to scale with D = 40°, E = 65° and F = 75°; the longest side DE is opposite the largest angle F, and the shortest side EF is opposite the smallest angle D.

Triangle Inequality: Can These Sides Form a Triangle?

Three lengths form a triangle only if each pair adds to more than the third. A shortcut: check that the two shorter sides sum to more than the longest. Lengths 4, 6 and 9 work because

4 + 6 = 10

which beats 9. Lengths 3, 5 and 8 fail because 3 + 5 only equals 8, which lies flat as a segment.

Lengths 4, 6 and 9 form a triangle because 4 + 6 = 10 is more than 9; lengths 3 and 5 laid along 8 lie flat because 3 + 5 = 8, so no triangle forms.

Integer Side Constraints

If two sides are a and b with a smaller than b, the third side must be greater than b - a and less than a + b. With sides 6 and 11, the third side is between 5 and 17, not including either, so whole-number options run from 6 to 16.

Perimeter of a Triangle

The perimeter is the total distance around a triangle: add the three side lengths. A triangular garden bed with sides of 7 ft, 10 ft and 12 ft has a perimeter of 29 ft. Keep units consistent, converting inches and feet to one unit before adding.

Area: Half Base Times Perpendicular Height

The area of a triangle is

Here b is the length of a chosen base and h is the perpendicular height to that base, the distance from the opposite vertex measured along a line that meets the base at a right angle. Area is in square units. A triangle with base 14 m and height 9 m has area 63 square meters.

Triangle with base 14 m and a dashed perpendicular height of 9 m from the top vertex to the base; area = ½ × 14 × 9 = 63 square meters.

Area of Right and Obtuse Triangles

In a right triangle the two legs are perpendicular, so one leg serves as the base and the other as the height: legs of 8 and 15 give

square units. In an obtuse triangle, the height to a short base may fall outside the triangle. Extend the base with a dashed line and measure the perpendicular from the top vertex to that line; the same formula still works.

Left: right triangle with perpendicular legs 15 and 8, area ½(8)(15) = 60. Right: obtuse triangle whose base b is extended with a dashed line so the perpendicular height h falls outside the triangle.

Why the Area Formula Works

Two identical copies of any triangle can be fitted together to form a parallelogram with the same base and height. A parallelogram's area is base times height, so one triangle is exactly half of it. That explains the factor of one-half, and also why a slanted side cannot stand in for the height.

A triangle with base b and height h and an identical copy turned 180° fit together to form a parallelogram with the same base and height: parallelogram area = bh, so one triangle = ½bh.

Right Triangle Vocabulary

In a right triangle, the side opposite the 90-degree angle is the hypotenuse, usually labeled c, and it is always the longest side. The two sides that form the right angle are the legs, a and b. The two other angles are acute and complementary, adding to 90 degrees. The sides are linked by

Right triangle ABC with the right angle at C: legs a and b form the right angle, hypotenuse c is opposite the 90° angle and is the longest side, acute angles A + B = 90°, and a² + b² = c².

Pythagorean Theorem: Find Hypotenuse

To find the hypotenuse, square each leg, add, and take the square root. With legs 9 and 12:

Right triangle with legs 9 and 12 and hypotenuse 15, with a square on each side: areas 81 and 144 on the legs and 225 on the hypotenuse; 81 + 144 = 225, so c = √225 = 15.

Check that the answer is longer than either leg; a hypotenuse shorter than a leg signals an error.

Pythagorean Theorem: Find a Missing Leg

To find a missing leg, subtract the square of the known leg from the square of the hypotenuse. With hypotenuse 13 and leg 5:

Converse of the Pythagorean Theorem

With c the longest side, if

the triangle is right. If

it is acute, and if

it is obtuse. Sides 6, 7 and 10:

36 + 49 = 85

less than 100, so obtuse.

Triangle drawn to scale with sides 6, 7 and 10; the angle opposite the 10 side is obtuse because 36 + 49 = 85 is less than 10² = 100.

Recognize Useful Pythagorean Triples

Some whole-number side sets appear again and again: 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any whole-number multiple also works, such as 6-8-10, 9-12-15 or 15-20-25. Spotting a scaled triple, like legs of 30 and 40, gives the hypotenuse 50 instantly, with no square roots.

Congruent Triangles

Congruent triangles have exactly the same size and shape, so all matching sides and angles are equal. You do not need all six facts to prove congruence. Valid shortcuts are SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and a side not between them), and HL for right triangles (hypotenuse and one leg). SSA and AAA are not valid: SSA can fit two different triangles, and AAA fixes only shape, not size.

Congruence shortcuts, with the known parts highlighted: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them). More shortcuts: AAS (two angles and a side not between them) and HL (hypotenuse and a leg of a right triangle); SSA is not valid because the same side, side and angle fit two different triangles.

Similar Triangles

Similar triangles have the same shape but possibly different sizes. Matching angles are equal and matching sides share one ratio, the scale factor k. If a triangle has sides 4, 6 and 9 and a similar triangle has a side of 10 matching the 4, then

and its other sides are 15 and 22.5.

Similar triangles drawn to scale with the same angles: sides 4, 6 and 9, and an enlarged copy with sides 10, 15 and 22.5; matching sides share a color, equal angles have the same number of arcs, and k = 10 ÷ 4 = 2.5.

Similarity Criteria

Triangles are similar if two pairs of angles are equal (AA), if all three pairs of sides are in the same ratio (SSS similarity), or if two pairs of sides are in the same ratio and the angles between them are equal (SAS similarity). AA is the most common test, since matching two angles forces the third angles to match as well.

Similarity tests, each with a small triangle and a copy twice as large: AA (two pairs of equal angles, single and double arcs), SSS (sides 4, 6, 9 and 8, 12, 18, all sides × 2), SAS (sides 6 and 4 and 12 and 8 with the same angle between them).

Similarity and Perimeter Ratios

For similar triangles with scale factor k, every length scales by k, including perimeters and heights. A triangle with perimeter 24 enlarged by

has perimeter 36. Working the other way, the ratio of the perimeters gives the scale factor for the sides.

Similarity and Area Ratios

Area depends on two lengths, a base and a height, and each one is multiplied by k. So the areas of similar triangles scale by . If every side triples, the area becomes 9 times as large: a 5-square-inch triangle becomes 45 square inches. If two areas are in the ratio 16 to 25, the sides are in the ratio 4 to 5.

A 5 in² triangle and a copy with every side tripled; the large triangle is tiled by 9 copies of the small one, so its area is 9 × 5 = 45 in².

Special Right Triangle: 45°–45°–90°

A 45-45-90 triangle is an isosceles right triangle. Its legs are equal, and the hypotenuse is a leg times , so the sides are in the ratio . Legs of 5 give a hypotenuse of . A hypotenuse of 8 gives legs of

45-45-90 triangle with two equal legs of 5, a right-angle mark, two 45° angles and hypotenuse 5√2; side ratio 1 : 1 : √2.

Special Right Triangle: 30°–60°–90°

In a 30-60-90 triangle, the sides are in the ratio . The short leg, opposite 30 degrees, is half the hypotenuse; the long leg is the short leg times . A hypotenuse of 18 gives legs of 9 and .

30-60-90 triangle with hypotenuse 18, short leg 9 opposite the 30° angle and long leg 9√3; side ratio 1 : √3 : 2.

Angles From a Diagram: Facts Versus Appearance

Geometry figures on the test may not be drawn to scale. An angle that looks like 90 degrees is a right angle only if it is marked with a small square or stated in words. Sides that look equal are equal only with tick marks or given lengths. Base every conclusion on the numbers and markings provided, not on how the picture looks.

Left: a triangle whose angle looks like 90° and legs look equal but have no marks, so nothing can be assumed. Right: the same triangle with a right-angle square and tick marks, so it has a right angle and equal legs.

Combined Triangle Angle and Length Problems

Multi-step questions often move between angles and lengths. A right triangle with one acute angle of 30 degrees and a hypotenuse of 20 is a 30-60-90 triangle, so its legs are 10 and . Its area is then half of 10 times , or square units. Identify the triangle type first, then choose the length tool.

30-60-90 right triangle with hypotenuse 20 and a 30° angle; legs 10 (opposite 30°) and 10√3, area ½ × 10 × 10√3 = 50√3.

No-Calculator Triangle Toolkit

Keep a short toolkit. Angles: the three total 180, and an exterior angle equals the two remote interior angles. Sides: the two shorter sides must add to more than the longest. Right triangles: look for a known triple before squaring, and pull out common factors first; legs of 18 and 24 are 6 times 3 and 4, so the hypotenuse is

Similar figures: lengths scale by k, areas by .

Worked Example 1 — Unknown Algebraic Angles

Problem: A triangle's angles measure (2x + 5)°, (3x − 10)° and (x + 17)°. Find each angle.

Step 1: The three angles of any triangle add to 180°.

Step 2: Combine like terms. The x-terms:

2x + 3x + x = 6x

The numbers:

5 − 10 + 17 = 12

Step 3: Subtract 12 from both sides, then divide by 6.

Step 4: Substitute x = 28 into each expression.

Answer: The angles are 61°, 74° and 45°.

Check:

61 + 74 = 135
and
135 + 45 = 180

Worked Example 2 — Exterior Angle and Adjacent Interior

Problem: In triangle ABC, the exterior angle at C measures 128° and angle A is 51°. Find angle B and the interior angle C.

Triangle ABC with angle A = 51°, angle B = ? and interior angle C = ?; side AC is extended past C, making an exterior angle of 128° at C.

Step 1: An exterior angle of a triangle equals the sum of the two interior angles that are not next to it (the remote angles A and B).

Step 2: Put in A = 51 and subtract to find B.

Step 3: The exterior angle and interior angle C sit side by side on a straight line, so they add to 180°.

Answer: Angle B is 77° and interior angle C is 52°.

Check: The three interior angles must total 180°:

51 + 77 = 128
and
128 + 52 = 180

Worked Example 3 — Strict Side-Length Bounds

Problem: Two sides of a triangle are 8 and 13. Which whole numbers can the third side be, and how many are there?

Step 1: Triangle Inequality: the third side must be longer than the difference of the other two sides, or the two short sides could not meet.

Step 2: The third side must also be shorter than the sum of the other two sides.

Step 3: Combine the two limits. The third side s is strictly between 5 and 21.

Step 4: The whole numbers that fit are 6, 7, 8, and so on up to 20. Count them: last minus first, plus 1.

Number line from 0 to 22: open circles at 5 and 21 (excluded), the whole numbers 6 through 20 marked between them; 5 < s < 21 gives 15 whole numbers.

Answer: The third side can be any whole number from 6 to 20, which is 15 possible values.

Check: A side of 5 fails because

8 + 5 = 13

only equals 13 and does not exceed it. A side of 21 fails because

8 + 13 = 21

does not exceed 21. So 5 and 21 are both excluded.

Worked Example 4 — Isosceles Angles

Problem: An isosceles triangle has two base angles of (3x + 6)° each and a vertex angle of 2x°. Find all the angles.

Isosceles triangle with tick marks on the two equal sides, base angles labeled (3x + 6)° each and vertex angle labeled 2x°.

Step 1: The angles of a triangle add to 180°. There are two base angles, so count that expression twice.

Step 2: Distribute the 2: 2 times 3x is 6x, and 2 times 6 is 12.

Step 3: Combine like terms:

6x + 2x = 8x

Step 4: Subtract 12 from both sides, then divide by 8.

Step 5: Substitute x = 21.

Answer: Each base angle is 69° and the vertex angle is 42°.

Check:

69 + 69 + 42 = 138 + 42 = 180

Worked Example 5 — A Right Triangle With Missing Leg

Problem: A 26-foot ladder leans against a wall with its foot 10 feet from the wall. How high up the wall does it reach?

A 26 ft ladder leans against a wall with its foot 10 ft from the wall; the wall, ground and ladder form a right triangle, and the height h up the wall is unknown (h = ?).

Step 1: The wall, the ground and the ladder form a right triangle. The ladder is the hypotenuse c = 26, the ground distance is one leg a = 10, and the height h is the other leg. Use the Pythagorean theorem.

Step 2: Square the known numbers.

Step 3: Subtract 100 from both sides.

Step 4: Take the square root. Since

24 × 24 = 576

the square root is 24.

Answer: The ladder reaches 24 feet up the wall.

Check: 10² + 24² =

100 + 576 = 676 = 26

². This is the 5-12-13 triangle with every side doubled.

Worked Example 6 — Special Right Triangles

Problem: (a) A square has a diagonal of 7√2 cm. Find its side. (b) A 30-60-90 triangle has a long leg of 6√3. Find its short leg and hypotenuse.

(a) A square with side s = ? and diagonal 7√2 cm, which cuts it into two 45-45-90 triangles. (b) A 30-60-90 triangle with long leg 6√3 opposite the 60° angle; the short leg and hypotenuse are marked ?.

Step 1: The diagonal cuts the square into two 45-45-90 triangles. In a 45-45-90 triangle, the hypotenuse is the leg times √2.

Step 2: The diagonal is the hypotenuse. Match 7√2 with leg × √2, so the leg (the side of the square) is 7.

Step 3: In a 30-60-90 triangle, the sides are short leg, short leg × √3, and 2 × short leg. The long leg is short leg × √3.

So the short leg is 6.

Step 4: The hypotenuse is twice the short leg.

Answer: (a) Each side of the square is 7 cm. (b) The short leg is 6 and the hypotenuse is 12.

Check: For (b), 6² + (6√3)² =

36 + 36 × 3 = 36 + 108 = 144 = 12

². For (a), 7² + 7² = 98, and (7√2)² =

49 × 2 = 98

Worked Example 7 — Similar Triangles

Problem: A 6-foot post casts a 4-foot shadow while a nearby tree casts a 22-foot shadow. How tall is the tree?

A 6 ft post with a 4 ft shadow and a tree of unknown height h with a 22 ft shadow, drawn to scale; parallel sun rays make the same angle with the ground at both shadow tips, and both objects stand at right angles to the ground.

Step 1: The sun hits both objects at the same angle, so the post with its shadow and the tree with its shadow form similar triangles. Their sides are in the same ratio: height over shadow.

Step 2: Simplify 6/4. Dividing 6 by 4 gives 1.5, so each object is 1.5 times as tall as its shadow.

Step 3: Multiply both sides by 22 to get h alone.

Answer: The tree is 33 feet tall.

Check: Another way: the tree's shadow is

22 ÷ 4 = 5.5

times the post's shadow, so the tree is

5.5 × 6 = 33 feet

Same answer.

Worked Example 8 — Area and Perpendicular Height

Problem: Triangle PQR has base PQ = 12 cm. Side PR is 10 cm long, and the perpendicular distance from R to line PQ is 7 cm. Find the area.

Triangle PQR with base PQ = 12 cm, slanted side PR = 10 cm and a dashed perpendicular height of 7 cm from R to PQ; the area is unknown (A = ?).

Step 1: Use the triangle area formula. The height must be perpendicular (at a right angle) to the base.

Step 2: Choose the right numbers. The base is 12. The height is the perpendicular distance, 7. Side PR (10 cm) is slanted, so it is not the height and does not go in the formula.

Step 3: Substitute and multiply. Half of 12 is 6, which keeps the arithmetic easy.

Answer: The area is 42 square centimeters.

Check: A rectangle 12 by 7 has area 84, and a triangle with the same base and height is half of that:

84 ÷ 2 = 42
Check for triangle PQR: a 12 by 7 rectangle around the triangle has area 12 × 7 = 84 cm², and the triangle with the same base and height is half: 84 ÷ 2 = 42 cm².

Worked Example 9 — Similarity Changes Area

Problem: A triangle with an area of 20 square meters is enlarged so every side is 3 times as long. Find the new area.

Step 1: Area uses two lengths multiplied together (base and height). Both of them get 3 times longer, so the area is multiplied by 3 twice.

Step 2: Multiply the original area by this area scale factor.

Answer: The new area is 180 square meters, not 60.

Check: Suppose the original has base 10 and height 4:

1/2 × 10 × 4 = 20

Enlarged, base 30 and height 12:

1/2 × 30 × 12 = 180

It matches. Multiplying by 3 alone (to get 60) is the common trap.

High-Value Triangle Traps

Using a slanted side as the height: the height must be perpendicular to the base. Forgetting the one-half in , which doubles the answer. Adding the legs instead of their squares in the Pythagorean theorem, or adding squares when finding a missing leg instead of subtracting. Scaling area by k instead of . Accepting SSA or AAA as a congruence test. Allowing a third side equal to the sum or difference of the other two; the inequality is strict. Mixing up 30-60-90 legs: the short leg is half the hypotenuse, and belongs to the long leg.

Triangle Formula and Reasoning Review

Interior angles total 180 degrees, an exterior angle equals the sum of the two remote interior angles, and isosceles base angles are equal. The longest side faces the largest angle, and any side must be shorter than the sum and longer than the difference of the other two. Perimeter adds the sides; area is with a perpendicular height. For right triangles,

and the triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25 save time. Special ratios are and . Similar triangles scale lengths by k and areas by .

Quick review

Triangle angle sum and the exterior angle theorem?
The three interior angles total 180°. Each exterior angle equals the two non-adjacent interior angles added together. With remote angles 45° and 70°, the exterior angle is 115° and the adjacent interior angle is 65°.
Triangle inequality: can sides 5, 7 and 13 form a triangle?
No. The sum of the two shorter sides has to exceed the longest side, but 5 + 7 = 12 < 13. With two sides a < b, the third side lies strictly between b − a and a + b.
Area of a triangle?
, where b is the base and h is the height perpendicular to that base (not a slanted side); the answer is in square units. Base 10 cm and height 6 cm give 30 cm².
Pythagorean theorem?
In a right triangle, , with legs a and b and hypotenuse c (the longest side, opposite the right angle). Legs 6 and 8 give . For a missing leg, subtract the squares.
Which Pythagorean triples should you memorize?
3-4-5, 5-12-13, 8-15-17 and 7-24-25, plus any whole-number multiple of them, such as 6-8-10 or 10-24-26. Spotting one gives the missing side without square roots.
Side ratios of the special right triangles?
45-45-90: , so hypotenuse = leg . 30-60-90: ; the short leg is half the hypotenuse and the long leg is short leg . Hypotenuse 14: legs 7 and .
Similar triangles: how do lengths and areas scale?
With scale factor k, every length (sides, perimeter, heights) is multiplied by k, but area is multiplied by . Doubling every side of a 12 cm² triangle gives 48 cm², not 24 cm².
Which triangle congruence shortcuts are valid?
Valid: SSS, SAS, ASA, AAS, and HL for right triangles. Not valid: SSA, which can fit two different triangles, and AAA, which fixes the shape but not the size.
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About this lesson

You will apply the triangle inequality and integer side constraints, find perimeter and area using half base times perpendicular height, and see why the area formula works. The right-triangle sections teach the Pythagorean theorem for a hypotenuse or a missing leg, its converse and useful Pythagorean triples. Congruence, similarity criteria, perimeter and area ratios, and the 45-45-90 and 30-60-90 special right triangles follow, along with a reminder to trust stated facts over how a diagram looks.

Nine worked examples and a set of high-value traps complete the lesson. This material supports Mathematics Knowledge, an AFQT subtest. Allow about 43 minutes, then take the 20-question lesson test.

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