GEO05-1 Angles Of Triangles

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Angles of Triangles

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Given: △ABC
Prove: m∠A + m∠B + m∠C = 180°

Proof:

Learn
Interior Angles of Triangles

Step 3

Step 2

Step 1

Step 4

Theorem 5.1: Triangle Angle-Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°.

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Angles of Triangles

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Learn
Exterior Angles of Triangles

exterior
angles

An exterior angle of a triangle is an
angle formed by one side of the triangle
and the extension of an adjacent side. A
triangle has three exterior angles. ∠4 is
an exterior angle of △ABC.

remote
interior
angles

Each exterior angle of a triangle has two
remote interior angles that are not
adjacent to the exterior angle. ∠1 and ∠3
are the remote interior angles for ∠4.

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Angles of Triangles

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Learn
Exterior Angles of Triangles

Theorem 5.2: Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior
angles.

Given: △ABC

Prove: m∠A + m∠B = m∠1

(continued on the next slide)

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Angles of Triangles

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Example 2
Use the Exterior Angle Theorem

ARCHITECTURE Find the measure of
∠DAB in the front face of the building.

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Angles of Triangles

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Example 2
Use the Exterior Angle Theorem

m∠DAB = m∠ABC + m∠BCA

Exterior Angle
Theorem

(12x + 7)° = (6x − 4)° + 65°

Substitution

x = 9

Solve.

m∠DAB = [12(9) + 7]° or 115°

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Angles of Triangles

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Example 2
Use the Exterior Angle Theorem

Check

PUZZLES Find the measure of ∠XYZ.

90°

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Angles of Triangles

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Learn
Triangle Angle-Sum Corollaries

A corollary is a theorem with a proof that follows as a direct result
of another theorem. As with a theorem, a corollary can be used as a
reason in a proof. The corollaries below follow directly from the
Triangle Angle-Sum Theorem. (basically postulates)

Corollary 5.1
The acute angles of a right triangle are complementary.

Corollary 5.2
There can be at most one right or obtuse angle in a triangle.

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Angles of Triangles

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Example 3
Find Angle Measures in Right Triangles

Find each measure.

a. m∠BCD

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Angles of Triangles

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Example 3
Find Angle Measures in Right Triangles

m∠BCD = [2(28) − 2]° or 54°

m∠BCD + m∠DBC = 90°

Corollary 5.1

(x + 8)° + (2x − 2)° = 90°

Substitution

x = 28

Solve.