Question

a. informally prove that pairs of triangles are similar. each pair must meet the requirements of one of the similar triangle theorems. certain problems also require the application of angle theorems
b. determine the values of the required sides or angles
figure 23 - 38
details

1. right angled triangle a c b, where d b is the altitude from a b. angle a c d, 59 degrees; angle a b c, angle 1; angle b a c, angle 2; angle d c b, angle 3. 8. triangles f g k and triangle m j h are connected by line segments k m. side lengths of triangle f g k: g f, 9.92 centimeters; g k, 7.2 centimeters. side lengths of triangles m j h: m h, 5.5 centimeters; j h, 6.2 centimeters.

7.
determine:
∠1
∠2
∠3

Explanation:

Step1: Find \(\angle2\)

In right - triangle \(ACD\), \(\angle ADC = 90^{\circ}\). Using the angle - sum property of a triangle (\(\angle A+\angle ADC+\angle ACD = 180^{\circ}\)), and given \(\angle ACD=59^{\circ}\).
\(\angle2=180^{\circ}-\angle ADC - \angle ACD\)
\(\angle2 = 180^{\circ}-90^{\circ}-59^{\circ}=31^{\circ}\)

Step2: Find \(\angle1\)

In right - triangle \(ACB\), \(\angle ACB = 90^{\circ}\). Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle ACB=180^{\circ}\)), and we know \(\angle A=\angle2 = 31^{\circ}\)
\(\angle1=180^{\circ}-\angle ACB-\angle2\)
\(\angle1=180^{\circ}-90^{\circ}-31^{\circ}=59^{\circ}\)

Step3: Find \(\angle3\)

In right - triangle \(BCD\), \(\angle BDC = 90^{\circ}\). Using the angle - sum property of a triangle (\(\angle B+\angle BDC+\angle3 = 180^{\circ}\)), and \(\angle B=\angle1 = 59^{\circ}\)
\(\angle3=180^{\circ}-\angle BDC-\angle1\)
\(\angle3=180^{\circ}-90^{\circ}-59^{\circ}=31^{\circ}\)

Answer:

\(\angle1 = 59^{\circ}\), \(\angle2=31^{\circ}\), \(\angle3 = 31^{\circ}\)