DWG NO. 01.3 — Lesson 3 of 5

Angles and Angle Measures

Unit 1: Foundations of Geometry · ~15–30 min

Objective

Name angles correctly, classify them by measure, and use the Angle Addition Postulate to find unknown angle measures.

What is an angle?

An angle is formed by two rays that share a common endpoint, called the vertex. The rays are its sides. An angle can be named by its vertex letter alone if there's no ambiguity (∠B), by three points with the vertex in the middle (∠ABC or ∠CBA), or by a number (∠1).

Measuring angles — the Protractor Postulate

Angles are measured in degrees, written m∠ABC. The Protractor Postulate guarantees that the rays from a point can be matched to the numbers 0–180 so that an angle's measure equals the difference between the two numbers its sides land on — which is exactly how a physical protractor works.

Classifying angles by measure

Type	Measure
Acute	Between 0° and 90°
Right	Exactly 90° (marked with a small square)
Obtuse	Between 90° and 180°
Straight	Exactly 180° (the two sides form a line)

The Angle Addition Postulate

If ray BD lies in the interior of ∠ABC, then the two smaller angles add up to the whole: m∠ABD + m∠DBC = m∠ABC.

Ray BD splits ∠ABC into ∠ABD and ∠DBC.

Worked Example 1 · Angle Addition Postulate

ProblemRay BD lies in the interior of ∠ABC. m∠ABD = 35° and m∠DBC = 48°. Find m∠ABC and classify it.

1.m∠ABC = m∠ABD + m∠DBC = 35° + 48°

2.m∠ABC = 83°, which is between 0° and 90°

m∠ABC = 83° — acute

Worked Example 2 · Solving algebraically

Problemm∠XYZ = 112°. Ray YW splits it so that m∠XYW = (3x+10)° and m∠WYZ = (2x+2)°. Find x and both angle measures.

1.(3x+10) + (2x+2) = 112

2.5x + 12 = 112 → 5x = 100 → x = 20

3.m∠XYW = 3(20)+10 = 70°. m∠WYZ = 2(20)+2 = 42°. Check: 70 + 42 = 112 ✓

x = 20 • m∠XYW = 70° • m∠WYZ = 42° • ∠XYZ is obtuse

Guided practice

1.Classify an angle that measures 90°.
Show answer
Right angle.
2.Classify an angle that measures 172°.
Show answer
Obtuse angle (between 90° and 180°).
3.m∠PQR = 64°. Ray QT is in its interior with m∠PQT = 27°. Find m∠TQR.
Show answer
m∠PQT + m∠TQR = m∠PQR → 27 + m∠TQR = 64 → m∠TQR = 37°
4.m∠ABC = 150°. Ray BD is in its interior with m∠ABD = (4x+6)° and m∠DBC = (x+9)°. Find x and m∠ABD.
Show answer
4x+6+x+9 = 150 → 5x+15 = 150 → 5x = 135 → x = 27. m∠ABD = 4(27)+6 = 114°
5.Given vertex B with rays to points A and C, give two correct names for the angle.
Show answer
∠ABC or ∠CBA (and ∠B, if there's only one angle at vertex B in the figure).

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