8-2 The Triangle Sum Theorem notes

Mrs. Aitken’s Integrated 2
Unit 8 Similar and Congruent Triangles
8-2 The Triangle Sum Theorem
Warm-up
1. What is the degree measure of

Mrs. Aitken’s Integrated 2
Unit 8 Similar and Congruent Triangles
Example 1
Write a two column proof of the Triangle Sum Theorem.
Given: ▲ XYZ
Prove: m∠1 + m∠2 + m∠3 = 180°
Solution 1
Plan ahead - draw a helping line through Z parallel to XY. Then use the whole
and parts postulate and alternate interior angles to show that the sum of
the angles of a triangle equals the sum of the measures of the parts of a
straight angle.
Statements
Justifications
1. ▲ XYZ
1. Given
2. Draw a helping line PQ parallel
to XY through vertex Z.
2. Through a point not on a given line,
there one and only one line parallel
to the given line.
3. m∠PZQ = 180° 3. Straight Angle postulate
4. m∠PZQ = m∠4 + m∠3 + m∠5 4. Whole and parts postulate
5. m∠4 + m∠3 + m∠5 = 180° 5. Substitution property of equality
6. m∠1 = m∠4 and m∠2 = m∠5 6. If 2 parallel lines are cut by a
transversal, then alternate
interior angles are equal in
measure
7. m∠1 + m∠2 + m∠3 = = 180° 7. Substitution property of equality
8-2 The Triangle Sum Theorem

Mrs. Aitken’s Integrated 2
Unit 8 Similar and Congruent Triangles
Example 2
Find each unknown angle measure.
Solution 2
2x + x + x + 20 = 180
4x = 160
x = 40
2x°
80°
(x + 20)° x°
60° 40°
Quadrilateral Theorems
Theorem - The sum of the measures of the angles of a (convex) quadrilateral
is 360°.
Look at proof on page 441 - Sample 2
m∠1 + m∠2 + m∠3 + m∠4 = 360°
Theorem - If both pairs of opposite angles of a quadrilateral are equal in
measure, than the quadrilateral is a parallelogram
- an extension of the theorems we already learned.
implies
Example 3
In quadrilateral ABCD, m∠A = m∠C = x° and m∠B = m∠D = 2x°
Solution 3
x + x + 2x + 2x = 360
6x = 360
x = 60
m∠A = m∠C = 60° and m∠B = m∠D = 120°
8-2 The Triangle Sum Theorem

Mrs. Aitken’s Integrated 2
Unit 8 Similar and Congruent Triangles
Key Term
Remote interior angles - The two angles of a triangle that are not next to a
given exterior angle of the triangle.
F
C exterior angle
Remote
interior angle
B
D
A
Remote
interior angle
E
Exterior Angle Theorem
Theorem - The measure of an exterior angle of a triangle is equal to the sum
of the measures of the two remote interior angles. Can be proven with the
Triangle Sum Theorem
F
C
exterior angle
exterior angle
B
D
A
exterior angle
E
Example 4 - Find the unknown angle measure
Solution 4
21 + 50 + x = 180
x = 109°
180 - x = y
180 - 109 = y
y = 71°
180 - 50 = z
z = 130°
x°
y°
21° 50° z°
8-2 The Triangle Sum Theorem