Geometry

Given m || n, find the value of x and y. The angles formed by a transversal cutting parallel lines m and n are: At line n: (y+16)°, (3x-15)°, and (2x+7)°

Step-by-step solution with explanation

Final Answer

and

Step-by-step solution

Identify angle relationships at line n

All three angles are formed at line n by the transversal. We need to figure out how they relate to each other — whether they are supplementary, vertical, or on a straight line — to set up our equations.

Set linear pair equal to 180 degrees

Two of the angles form a linear pair (they are adjacent and together make a straight line). A straight line always measures 180°, so their sum equals 180°.

Solve the equation for x

Combine like terms on the left side, then add 8 to both sides, and finally divide by 5 to isolate x.

Find the measure of (2x+7)°

Plug x = 37.6 into the expression (2x+7) to find the actual angle measure. We'll use this to find y.

Set vertical angles equal to find y

The angle (y+16)° is a vertical angle with (2x+7)°. Vertical angles are always equal, so we set them equal to each other.

Solve for y

Subtract 16 from both sides to isolate y. This gives us the value of y.

Understanding this problem

Learning Insight

When a transversal crosses parallel lines, it creates special angle pairs — vertical angles (equal) and linear pairs (summing to 180°). These relationships come from the Parallel Lines Postulate and the Straight Angle Theorem, and they always hold true no matter the angle measure.

Quick Tip

Look for a straight line first — any two angles that sit side by side on a straight line must add to 180°. Then look for vertical angles (the X shape) — they are always equal. Spotting these two patterns solves most transversal problems fast.

Common Mistake

Students often add all three angles together and set them equal to 180°, treating them as if they all lie on one straight line. Only the two angles that actually form a linear pair should be added to get 180°.