Solved Find each angle measure: 1. 2. 义 670 1420 m/1. GHJK ... - Free Printable

Let's solve each problem step by step using geometric principles such as vertical angles, supplementary angles, corresponding angles, and parallel lines.

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1. Find m∠1

We are given two intersecting lines forming an "X" shape, with one angle labeled $67^\circ$. The angle marked $ \angle 1 $ is opposite to the $67^\circ$ angle.

- Vertical angles are equal.
- So, $ \angle 1 $ is vertical to the $67^\circ$ angle.

✔ Therefore:
$$
m\angle 1 = 67^\circ
$$

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2. Find m∠HJK

We have two parallel lines cut by a transversal. One angle is $142^\circ$, and we are to find $x^\circ = \angle HJK$.

Looking at the diagram:
- The $142^\circ$ angle and $\angle HJK$ are on the same side of the transversal and between the two parallel lines — so they are consecutive interior angles (same-side interior angles).
- Consecutive interior angles are supplementary, meaning they add up to $180^\circ$.

So:
$$
x + 142^\circ = 180^\circ \\
x = 180^\circ - 142^\circ = 38^\circ
$$

✔ Therefore:
$$
m\angle HJK = 38^\circ
$$

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3. Find m∠3

Two parallel lines are cut by a transversal. One angle is $111^\circ$, and we are to find $ \angle 3 $.

From the diagram:
- $ \angle 3 $ is adjacent to the $111^\circ$ angle on a straight line — so they form a linear pair.
- Linear pairs are supplementary: they add to $180^\circ$.

So:
$$
m\angle 3 + 111^\circ = 180^\circ \\
m\angle 3 = 180^\circ - 111^\circ = 69^\circ
$$

✔ Therefore:
$$
m\angle 3 = 69^\circ
$$

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4. Find m∠4

We have two lines intersecting, and there’s a right angle symbol at the intersection.

This means that the angle formed is $90^\circ$. Since $ \angle 4 $ is adjacent to this right angle and shares a straight line, it must also be part of a linear pair.

But wait — actually, if the lines are perpendicular (indicated by the right angle), then all four angles formed are $90^\circ$.

So:
$$
m\angle 4 = 90^\circ
$$

✔ Therefore:
$$
m\angle 4 = 90^\circ
$$

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5. Find m∠BST

We have two parallel lines cut by a transversal. One angle is $138^\circ$, and we are to find $ \angle BST $, which is labeled $x^\circ$.

From the diagram:
- The $138^\circ$ angle and $ \angle BST $ are alternate interior angles (they are on opposite sides of the transversal and inside the parallel lines).
- Alternate interior angles are equal when lines are parallel.

So:
$$
m\angle BST = 138^\circ
$$

✔ Therefore:
$$
m\angle BST = 138^\circ
$$

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✔ Final Answers:

1. $ m\angle 1 = 67^\circ $
2. $ m\angle HJK = 38^\circ $
3. $ m\angle 3 = 69^\circ $
4. $ m\angle 4 = 90^\circ $
5. $ m\angle BST = 138^\circ $

Let me know if you'd like diagrams or further explanation!