parallel lines and triangles
find ( mangle 1 ).
1.
2.
3.
algebra find the value of each variable.
4.
5.
6.
7. use the diagram at the right to answer the questions.
a. which angle is an exterior angle?
b. what are its remote interior angles?
c. find ( mangle 1 ) and ( mangle 2 ).
find the value of the variables and the measures of the angles.
8.
9.
10. in the figure at the right, ( overline{ed} perp overline{cb} ) and ( overline{ed} ) bisects ( angle cdb ). find ( mangle dba ).
11. reasoning what is the measure of each angle in an isosceles right triangle? explain.
12. the ratio of the angle measures of the acute angles in a right triangle is ( 2:3 ). find the measures of the acute angles.
prentice hall gold geometry • practice and problem solving workbook
copyright © savvas learning company llc. all rights reserved.
75

Question

parallel lines and triangles
find ( mangle 1 ).
1.
2.
3.
algebra find the value of each variable.
4.
5.
6.
7. use the diagram at the right to answer the questions.
   a. which angle is an exterior angle?
   b. what are its remote interior angles?
   c. find ( mangle 1 ) and ( mangle 2 ).
find the value of the variables and the measures of the angles.
8.
9.
10. in the figure at the right, ( overline{ed} perp overline{cb} ) and ( overline{ed} ) bisects ( angle cdb ). find ( mangle dba ).
11. reasoning what is the measure of each angle in an isosceles right triangle? explain.
12. the ratio of the angle measures of the acute angles in a right triangle is ( 2:3 ). find the measures of the acute angles.
prentice hall gold geometry • practice and problem solving workbook
copyright © savvas learning company llc. all rights reserved.
75

Accepted Answer

### Problem 1: Find $ m\angle 1 $ in the first triangle (with angles $ 92^\circ $ and $ 63^\circ $)
# Explanation:
## Step 1: Recall the triangle angle sum theorem
The sum of the interior angles of a triangle is $ 180^\circ $. So, if we have two angles, we can find the third by subtracting the sum of the known angles from $ 180^\circ $.
## Step 2: Calculate the sum of the known angles
The known angles are $ 92^\circ $ and $ 63^\circ $. Their sum is $ 92^\circ + 63^\circ = 155^\circ $.
## Step 3: Find $ m\angle 1 $
Subtract the sum of the known angles from $ 180^\circ $: $ m\angle 1 = 180^\circ - 155^\circ = 25^\circ $.
# Answer:
$ 25^\circ $

### Problem 2: Find $ m\angle 1 $ in the second triangle (with angles $ 20^\circ $ and $ 87^\circ $)
# Explanation:
## Step 1: Use the triangle angle sum theorem
The sum of interior angles in a triangle is $ 180^\circ $.
## Step 2: Sum the known angles
The known angles are $ 20^\circ $ and $ 87^\circ $. Their sum is $ 20^\circ + 87^\circ = 107^\circ $.
## Step 3: Calculate $ m\angle 1 $
Subtract from $ 180^\circ $: $ m\angle 1 = 180^\circ - 107^\circ = 73^\circ $.
# Answer:
$ 73^\circ $

### Problem 3: Find $ m\angle 1 $ in the third triangle (with angles $ 80^\circ $ and $ 40^\circ $)
# Explanation:
## Step 1: Apply the triangle angle sum theorem
The sum of interior angles of a triangle is $ 180^\circ $.
## Step 2: Sum the known angles
The known angles are $ 80^\circ $ and $ 40^\circ $. Their sum is $ 80^\circ + 40^\circ = 120^\circ $.
## Step 3: Find $ m\angle 1 $
Subtract from $ 180^\circ $: $ m\angle 1 = 180^\circ - 120^\circ = 60^\circ $.
# Answer:
$ 60^\circ $

### Problem 12: The ratio of the angle measures of the acute angles in a right triangle is $ 2:3 $. Find the measures of the acute angles.
# Explanation:
## Step 1: Recall properties of a right triangle
A right triangle has one $ 90^\circ $ angle, and the sum of the other two (acute) angles is $ 90^\circ $ (since the total sum of angles in a triangle is $ 180^\circ $, and $ 180^\circ - 90^\circ = 90^\circ $).
## Step 2: Let the measures of the acute angles be $ 2x $ and $ 3x $
Since their ratio is $ 2:3 $, we can represent them as $ 2x $ and $ 3x $.
## Step 3: Set up an equation
The sum of the acute angles is $ 90^\circ $, so $ 2x + 3x = 90^\circ $.
## Step 4: Solve for $ x $
Combine like terms: $ 5x = 90^\circ $. Then, $ x = \frac{90^\circ}{5} = 18^\circ $.
## Step 5: Find the measures of the angles
For $ 2x $: $ 2(18^\circ) = 36^\circ $. For $ 3x $: $ 3(18^\circ) = 54^\circ $.
# Answer:
The measures of the acute angles are $ 36^\circ $ and $ 54^\circ $.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: Find \( m\angle 1 \) in the first triangle (with angles \( 92^\circ \) and \( 63^\circ \))

Step 1: Recall the triangle angle sum theorem

Step 2: Calculate the sum of the known angles

Step 3: Find \( m\angle 1 \)

Step 1: Use the triangle angle sum theorem

Step 2: Sum the known angles

Step 3: Calculate \( m\angle 1 \)

Step 1: Apply the triangle angle sum theorem

Step 2: Sum the known angles

Step 3: Find \( m\angle 1 \)

Answer:

Problem 2: Find \( m\angle 1 \) in the second triangle (with angles \( 20^\circ \) and \( 87^\circ \))