UNIT 9: POLYGONS, SIMILARITY AND CONGRUENCE

Lesson 1: Reviewing Angle Relationships

Review important angle facts and use them to calculate unknown angles.

Learning Objectives

By the end of this lesson, you should be able to:

  • identify acute, right, obtuse, straight and reflex angles;
  • recognise adjacent, complementary and supplementary angles;
  • use angles on a straight line and angles around a point;
  • use vertically opposite angle facts;
  • identify angle relationships formed by parallel lines and a transversal; and
  • calculate unknown angles and give a reason for each step.

1. What Is an Angle?

An angle is formed when two rays meet at a common endpoint called the vertex. Angles are measured in degrees, written with the symbol °.

55°
vertex
arm
arm

2. Types of Angles

Acute angle
Greater than 0° but less than 90°.
Example: 42°
Right angle
Exactly 90°.
It is often marked with a small square.
Obtuse angle
Greater than 90° but less than 180°.
Example: 125°
Straight angle
Exactly 180°.
Its arms form a straight line.
Reflex angle
Greater than 180° but less than 360°.
Example: 240°
Complete turn
Exactly 360°.
It makes one full rotation.

3. Adjacent, Complementary and Supplementary Angles

Adjacent angles are next to each other. They have the same vertex and share one arm, but they do not overlap.

Complementary angles
Their measures add to 90°.
If one angle is 34°, the other is 90° − 34° = 56°.
Supplementary angles
Their measures add to 180°.
If one angle is 117°, the other is 180° − 117° = 63°.

Remember: complementary angles do not have to be adjacent. Supplementary angles do not have to form a straight line unless the diagram shows that they do.

4. Angles on a Straight Line

Angles that meet on a straight line add to 180°.

125°
x°
Worked Example 1
125° + x = 180°
x = 180° − 125°
x = 55°

5. Angles Around a Point

All angles around one point add to 360°.

Worked Example 2
Three angles around a point are 95°, 120° and x°.
95° + 120° + x = 360°
215° + x = 360°
x = 360° − 215°
x = 145°

6. Vertically Opposite Angles

When two straight lines cross, the angles directly opposite each other are called vertically opposite angles. They are always equal.

68°
x°

Worked Example 3
x and 68° are vertically opposite.
Therefore, x = 68°.
Each neighbouring angle is 180° − 68° = 112° because adjacent angles on the straight line are supplementary.

7. Parallel Lines and a Transversal

A transversal is a line that crosses two or more lines. When it crosses parallel lines, special angle relationships are formed.

Corresponding angles
They occupy matching corners at the two intersections.
They are equal.
Alternate angles
They lie between the parallel lines on opposite sides of the transversal.
They are equal.
Co-interior angles
They lie between the parallel lines on the same side of the transversal.
They add to 180°.
72°
x°
ℓ₁
ℓ₂
transversal
Worked Example 4
If ℓ₁ ∥ ℓ₂, the angles marked 72° and x° are corresponding angles.
Corresponding angles between parallel lines are equal.
Therefore, x = 72°.

8. Multi-Step Angle Problems

Some diagrams require more than one angle fact. Work one relationship at a time and write the reason for every step.

Worked Example 5
Two parallel lines are crossed by a transversal. An angle is 118°, and x is the co-interior angle beside it.
x + 118° = 180°   (co-interior angles)
x = 180° − 118°
x = 62°
Worked Example 6
Two vertically opposite angles are (3x + 10)° and 85°.
3x + 10 = 85   (vertically opposite angles are equal)
3x = 75
x = 25
Notice that x is a number; the actual angle is 85°.

Common Mistakes to Avoid

  • Using 360° for angles on a straight line instead of 180°.
  • Assuming adjacent angles are equal; they are only next to each other.
  • Confusing corresponding angles with co-interior angles.
  • Using parallel-line angle facts when the lines are not marked or stated as parallel.
  • Forgetting to write the degree symbol for an angle measure.
  • Finding x correctly but confusing x with the full angle expression.

A Helpful Problem-Solving Method

  1. Look for straight lines, intersecting lines, a point, or parallel-line marks.
  2. Name the angle relationship you can use.
  3. Write an equation using 90°, 180°, 360° or equal angles.
  4. Solve the equation carefully.
  5. Check whether the answer is a sensible angle size.
  6. Write the answer with its degree symbol and give the reason.

Lesson Summary

  • Complementary angles add to 90°.
  • Supplementary angles and angles on a straight line add to 180°.
  • Angles around a point add to 360°.
  • Vertically opposite angles are equal.
  • For parallel lines, corresponding and alternate angles are equal.
  • Co-interior angles between parallel lines add to 180°.

Practice Questions

  1. Classify an angle measuring 37°.
  2. Classify an angle measuring 180°.
  3. Find the complement of 28°.
  4. Find the supplement of 146°.
  5. Two adjacent angles on a straight line are x° and 73°. Find x.
  6. Angles around a point are 80°, 115°, 95° and y°. Find y.
  7. One angle at the intersection of two lines is 64°. Find its vertically opposite angle.
  8. Find an angle adjacent to the 64° angle in Question 7.
  9. Two parallel lines are crossed by a transversal. One corresponding angle is 108°. Find the other corresponding angle.
  10. Two co-interior angles are 112° and x°. Find x.
  11. Vertically opposite angles are (2x + 7)° and 73°. Find x.
  12. Two complementary angles are (3x)° and 24°. Find x and the first angle.

Answers

1. Acute   
2. Straight   
3. 62°   
4. 34°   
5. 107°   
6. 70°   
7. 64°   
8. 116°   
9. 108°   
10. 68°   
11. x = 33   
12. x = 22; first angle = 66°