Training Geometry Angles, Lines, and Triangles
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Angles, Lines, and Triangles

20 min Geometry

Angles, Lines, and Triangles

Geometry is the study of shapes, sizes, and the properties of space. It begins with the simplest objects — points, lines, and angles — and builds up to the rich world of triangles, polygons, circles, and solids.

Triangles are the workhorses of geometry. Every polygon can be decomposed into triangles, every surface can be approximated by triangular meshes, and many of the most powerful theorems — from Pythagoras to the law of cosines — are triangle theorems.

This lesson introduces angles, parallel and perpendicular lines, the angle relationships created by a transversal, and the fundamental properties of triangles including the triangle angle sum theorem.

Angle Classifications

TypeMeasure
Acute$0° < \theta < 90°$
Right$\theta = 90°$
Obtuse$90° < \theta < 180°$
Straight$\theta = 180°$
Definitions
  • Complementary angles: two angles whose sum is $90°$.
  • Supplementary angles: two angles whose sum is $180°$.
  • Vertical angles: when two lines intersect, opposite angles are equal.
Example 1

The complement of an angle is $27°$. Find the angle.

$\theta + 27° = 90° \implies \theta = 63°$.

Example 2

Two supplementary angles are in ratio $2:3$. Find them.

$2x + 3x = 180° \implies 5x = 180° \implies x = 36°$.

Angles: $72°$ and $108°$.

Triangles

Triangle Angle Sum

The sum of the interior angles of any triangle is $180°$.

By SidesBy Angles
Equilateral: 3 equal sides, 3 equal angles ($60°$)Acute: all angles $< 90°$
Isosceles: 2 equal sides, 2 equal base anglesRight: one $90°$ angle
Scalene: no equal sidesObtuse: one angle $> 90°$
Example 3

A triangle has angles $50°$ and $65°$. Find the third angle.

$180° - 50° - 65° = 65°$. (Isosceles — two equal angles.)

Pythagorean Theorem

Pythagorean Theorem

In a right triangle with legs $a, b$ and hypotenuse $c$:

$$a^2 + b^2 = c^2$$

Example 4

A right triangle has legs 6 and 8. Find the hypotenuse.

$$c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

Example 5

The hypotenuse is 13 and one leg is 5. Find the other leg.

$$b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$$

Example 6

Is a triangle with sides 7, 24, 25 a right triangle?

$7^2 + 24^2 = 49 + 576 = 625 = 25^2$ ✓ Yes.

Practice Problems

1. Find the supplement of $115°$.
2. Find the complement of $42°$.
3. A triangle has angles $35°$ and $90°$. Find the third.
4. Legs 9 and 12. Find the hypotenuse.
5. Hypotenuse 17, one leg 8. Find the other leg.
6. Is 5, 12, 13 a Pythagorean triple?
7. Two vertical angles: one is $4x + 10$, the other is $6x - 20$. Find $x$.
8. An isosceles triangle has a vertex angle of $40°$. Find each base angle.
9. Legs 1 and 1. Find the hypotenuse.
10. A 20-ft ladder leans against a wall, with its base 12 ft from the wall. How high does it reach?
11. Find the diagonal of a rectangle with sides 5 and 12.
12. Is 8, 15, 17 a right triangle?
13. One angle of a triangle is twice the smallest. The third is three times the smallest. Find all angles.
14. A guy wire is attached 40 ft up a pole. It anchors 30 ft from the base. How long is the wire?
15. Two supplementary angles differ by $36°$. Find both.
Show Answer Key

1. $65°$

2. $48°$

3. $55°$

4. $15$

5. $15$

6. Yes: $25 + 144 = 169$

7. $4x + 10 = 6x - 20 \implies x = 15$

8. $\frac{180° - 40°}{2} = 70°$ each

9. $\sqrt{2} \approx 1.414$

10. $\sqrt{400 - 144} = \sqrt{256} = 16$ ft

11. $\sqrt{25 + 144} = 13$

12. $64 + 225 = 289 = 17^2$ ✓ Yes

13. $x + 2x + 3x = 180° \implies x = 30°$. Angles: $30°, 60°, 90°$

14. $\sqrt{1600 + 900} = \sqrt{2500} = 50$ ft

15. $x + (x + 36) = 180 \implies x = 72$. Angles: $72°, 108°$

Perimeter and Area