Use the image to answer the question.

Question

An illustration shows a rectangle which is segmented left to right into a right triangle, a trapezoid in the middle, and another right triangle. The sides of the rectangle form the sides of the triangles. The top corners of the rectangle are denoted by two right angle symbols. The middle trapezoid is formed by two slanting lines from each of the top corners to the base of the rectangle. The angles formed by the two slanted lines, which become the hypotenuse of each right triangle, are labeled from 1 to 8 in clockwise direction starting from the top of the left side.

If m∠8=45° , what equation can you write to find the value of m∠7 ?

(1 point)

Answer 1

Given that \( m∠8 = 45° \), we can use the properties of angles in a triangle. Since \( m∠8 \) is the angle formed by the slanting line on the left triangle and the horizontal base, the two triangles formed on the left and right sides of the trapezoid are right triangles.

In a right triangle, the sum of the angles is always \( 180° \). Therefore, the angles in each right triangle consist of the right angle, the angle at the base (which is \( m∠7 \) in your description), and the angle on the hypotenuse (which is \( m∠8 \)).

For the left triangle: \[ m∠7 + m∠8 + 90° = 180° \]

Substituting \( m∠8 \) with \( 45° \): \[ m∠7 + 45° + 90° = 180° \]

This simplifies to: \[ m∠7 + 135° = 180° \]

Now, subtract \( 135° \) from both sides: \[ m∠7 = 180° - 135° \] \[ m∠7 = 45° \]

Thus, you can write the equation to find \( m∠7 \) as: \[ m∠7 + 45° + 90° = 180° \]