Find the measure of each angle indicated. Find the measure of the angle indicated in bold. Find the measure of each angle indicated. (3) (24) (85) Find the measure of the angle indicated in bold. (25) (26) Z + 4 17 + 1 X + T 1 + √2 #4 578
The Correct Answer and Explanation is:
Solutions
- Problem 3: The indicated angle is 57°.
- Problem 24: The indicated angle is 95°.
- Problem 25: The angle indicated in bold is 60°.
- Problem 26: The angle indicated in bold is 70°.
Explanations
Problem 3:
The diagram displays two parallel lines intersected by a third line, known as a transversal. The angle marked as 57° and the indicated angle are positioned as alternate interior angles. A fundamental rule in geometry states that when a transversal intersects parallel lines, the alternate interior angles are equal. Therefore, the measure of the indicated angle is also 57°.
Problem 24:
In this problem, two parallel lines are again cut by a transversal. The 85° angle and the angle marked with a question mark are consecutive interior angles, also called same-side interior angles. These angles are supplementary, meaning their sum is 180°. To find the measure of the unknown angle, subtract 85° from 180°. The calculation is 180° – 85° = 95°. The measure of the indicated angle is 95°.
Problem 25:
The objective is to find the measure of the bolded angle, represented by the expression (7x + 11). The two labeled angles, (7x + 11) and (17x + 1), are consecutive interior angles, so their measures sum to 180°. This allows for the creation of an algebraic equation to solve for x:
(7x + 11) + (17x + 1) = 180
Combine like terms: 24x + 12 = 180
Subtract 12 from both sides: 24x = 168
Divide by 24: x = 7
Now, substitute x = 7 into the expression for the bolded angle: 7(7) + 11 = 49 + 11 = 60.
The measure of the bolded angle is 60°.
Problem 26:
To determine the measure of the bolded angle, (x + 77), the value of x must be found first. The angles (x + 77) and (x + 117) are consecutive interior angles and are therefore supplementary. Their sum is 180°.
The equation is set up as follows:
(x + 77) + (x + 117) = 180
Combine like terms: 2x + 194 = 180
Subtract 194 from both sides: 2x = -14
Divide by 2: x = -7
Finally, substitute the value of x back into the expression for the bolded angle: (-7) + 77 = 70.
The measure of the angle indicated in bold is 70°.
