Download Scholarship Geometry Section 4-2: Angle Relationships in Triangles

Scholarship Geometry
Section 4-2: Angle Relationships in Triangles
Triangle Sum Theorem: The sum of the angle measures in a triangle is 180°.
Corollary: The acute angles of a right triangle are complementary (add to 90°).
B
m∠A + m∠B = 90°
A
C
Interior angles are angles inside a triangle.
Exterior angles are angles outside a triangle formed by extending one side of a triangle.
A Remote Exterior Angle is the exterior angle that is not adjacent to the two interior angles.
6
∠1, ∠2, ∠3 are interior angles
∠4, ∠5, ∠6 are exterior angles
∠4 is the remote exterior angle to ∠1, ∠2
∠5 is the remote exterior angle to ∠2, ∠3
∠6 is the remote exterior angle to ∠1, ∠3
2
1
3
4
5
Exterior Angle Theorem: The measure of an exterior angle is equal to the sum of the two
remote interior angles.
m∠4 = m∠1 + m∠2
2
1
3
4
Third Angles Theorem: If two angles of one triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
E
D
B
A
C
D
D
∠C ≅ ∠D
F
D
Ex. 1: Find the missing angles.
a) m∠XYZ
180 – (62 + 40) = 180 – 102 = 78°
b) m∠YWZ
180 – [(12 + 78) + 40] = 180 – (90 + 40)
= 180 – 130 = 50°
Ex. 2: Find y and find each angle.
The angles must sum to 180°.
(2y + 6) + (8y + 10) + (y + 32) = 180
11y + 48 = 180
11y = 132
y = 12
Angles:
2y + 6 = 2(12) + 6 = 30°
8y + 10 = 8(12) + 10 = 106°
y + 32 = 12 + 32 = 44°
Ex. 3: Find m∠ABD
The exterior angle = the sum of the two remote
interior angles:
2x + 16 = 58 + x + 12
2x + 16 = x + 70
x = 70 – 16 = 54
m∠ABD = 2x + 16 = 2(54) + 16 = 108 + 16 = 124°
Ex. 4: Find m∠N and m∠P
Since the other two angle pairs are congruent,
∠N and ∠P must also be congruent.
3x2 = x2 + 50
2x2 = 50
x2 = 25
m∠N = 3x2 = 3(25) = 75°
m∠P = x2 + 50 = 25 + 50 = 75°
Ex. 5: One acute angle of a right triangle is 62°. What is the other acute angle?
90 – 62 = 28°
Ex. 6: One acute angle of a right triangle is 2x. What is the other acute angle?
90 – 2x
Ex. 7: The three angles of a triangle are in the ratio of 2:3:5. Find the measure of each angle.
The three angles have measures 2x, 3x, and 5x, and they have to add to 180°:
2x + 3x + 5x = 180
10x = 180
x = 18
The angles are:
2x = 2(18) = 36°
3x = 3(18) = 54°
5x = 5(18) = 90°