Question

given that m || n and m∠1 = (5x + 20)° and m∠2 = (7x − 16)°, find m∠1. diagram of parallel lines m and n cut by a transversal, with angles 1 and 2 options: 15°, 110°, 94°, 18° question 19 5 pts which of the following could be the side lengths of a triangle? options: 5 in, 8 in, 10 in; 4 in, 6 in, 10 in; 1 in, 7 in, 9 in; 2 in, 3 in, 6 in

Explanation:

First Problem (Angle Calculation)

Step1: Identify Angle Relationship

Since \( m \parallel n \), \( \angle 1 \) and \( \angle 2 \) are alternate interior angles, so \( m\angle 1 = m\angle 2 \).

Step2: Set Up Equation

Set \( 5x + 20 = 7x - 16 \).

Step3: Solve for \( x \)

Subtract \( 5x \) from both sides: \( 20 = 2x - 16 \).
Add 16 to both sides: \( 36 = 2x \).
Divide by 2: \( x = 18 \).

Step4: Find \( m\angle 1 \)

Substitute \( x = 18 \) into \( 5x + 20 \): \( 5(18) + 20 = 90 + 20 = 110^\circ \).

Step1: Recall Triangle Inequality

For three lengths \( a, b, c \) (with \( c \) the largest), \( a + b > c \) must hold.

Step2: Check Each Option

• 5 in, 8 in, 10 in: \( 5 + 8 = 13 > 10 \), \( 5 + 10 = 15 > 8 \), \( 8 + 10 = 18 > 5 \) (valid).
• 4 in, 6 in, 10 in: \( 4 + 6 = 10 \), not \( > 10 \) (invalid).
• 1 in, 7 in, 9 in: \( 1 + 7 = 8 < 9 \) (invalid).
• 2 in, 3 in, 6 in: \( 2 + 3 = 5 < 6 \) (invalid).

Answer:

\( 110^\circ \) (corresponding to the option "110°")

Second Problem (Triangle Side Lengths)