Download Angles of Triangles Notes

Angles of Triangles
Name__________________________________
Through a point not on a line, there is one and only one line parallel to the given line.
P
Given ∆𝐴𝐵𝐶, construct a line through point B, parallel to ̅̅̅̅
𝐴𝐶 .
B
A
C
This auxiliary line (a line added to a diagram to help explain relationships in proofs) can be used
to explore the sum of the interior angles in a triangle.
Triangle Angle-Sum Theorem:
The sum of the measures of the angles of a triangle is 180°.
𝑚∠1 + 𝑚∠2 + 𝑚∠3 = 180°
1
2
3
Proof of Triangle Angle-Sum Theorem
P
B
1
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
A
Statements
2
R
3
C
Reasons
1. ABC , PR || AC
2. PBC and 3 are
supplementary
3. mPBC + m3 = 180°
4. mPBC = m1 + m2
5. m1 + m2 + m3 = 180°
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°
Exterior Angle of a Triangle: An exterior angle of a triangle is formed by
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exterior
angle
For each exterior angle, the two nonadjacent interior angles
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remote interior
angle
exterior
angle
remote interior
angle
Exterior Angle Theorem: The measure of each exterior angle of a triangle equals
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remote interior angle
remote interior angle
1
2
m4 = m1 + m2
3
4
Proof of Exterior Angle Theorem
1
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2
Statements
1. 4 is an exterior angle of the
triangle
2. 3 and 4 are supplementary
3. m3 + m4 = 180°
4. m1 + m2 + m3 = 180°
5. m3 + m4 = m1 + m2 + m3
6. m3 = m3
7. m4 = m1 + m2
exterior angle
Reasons
2
3
4
1. What are the values of x, y, and z?
x = 78, y = 102, z = 29
2. Find the measure of each angle.
50o
2x + 15
x + 25
75 degrees, 55 degrees
3. What is the measure of 1?
98 degrees
4. What is the measure of 2?
65 degrees
5. Two angles of a triangle measure 53°. What is the measure of an exterior angle at each
vertex of the triangle?
Each exterior angle is either 127 degrees or 106 degrees.
6. When radar tracks an object, the reflection of signals off the ground can result in clutter.
Clutter causes the receiver to confuse the real object with its reflection, called a ghost. In
the diagram, there is a radar receiver at A, an airplane at B, and the airplane’s ghost at D.
What is the value of x?
x = 50