Angles in Parallel Lines

The two marked angles occupy matching positions at the two crossings.

The marked angles are between the parallel lines and on opposite sides of the crossing line.

The marked angles are inside the parallel lines and on the same side of the crossing line.

Find the angle next to 63 degrees at the upper crossing, then match its position below.

The two marked angles occupy the same position at their crossings.

Turning the drawing does not change the corresponding-angle rule.

1. The marked angles are in matching positions at the two crossings.
2. Corresponding angles on parallel lines are equal, so x=48.

Locate the pair inside the parallel lines on opposite sides of the transversal.

1. The angles lie between the parallel lines on opposite sides of the transversal.
2. They are alternate angles, and alternate angles on parallel lines are equal.

A picture alone does not prove that two lines are parallel.

1. The corresponding-angle rule needs parallel lines.
2. The drawing does not state that these lines are parallel, so x cannot be determined from the information.

Use equal angles to find the lower acute angle, then use the straight line.

1. The angle above and right at the lower crossing is 71° (corresponding angles).
2. It and x lie on a straight line, so x=180-71=109.

The parallel-angle rules can also be used backwards to prove parallel lines.

1. Equal corresponding angles show that the two lines are parallel.
2. Equal angles do not show that the lines are perpendicular: that would require a right angle.

These co-interior angles total 180 degrees.

1. Co-interior angles on parallel lines add to 180°.
2. (3x+10)+80=180, so 3x+90=180.
3. 3x=90, so x=30.

Side AD is a transversal across the two parallel sides.

1. Angles A and D are inside the parallel sides on the same side of AD.
2. Co-interior angles on parallel lines add to 180°, so angle D =180-107=73°.

Adjacent angles total 180 degrees; use this twice.

Part (a)

1. AD is parallel to BC, so angles A and B are co-interior along AB.
2. Angle B =180-118=62°.

Part (b)

1. AB is parallel to DC, so angles B and C are co-interior along BC.
2. Angle C =180-62=118°. Opposite angles in the parallelogram are equal.

Transfer the 49-degree angle into the triangle first.

1. Angle ABC =49°, because alternate angles on parallel lines are equal.
2. Angles in triangle ABC add to 180°.
3. Angle BAC =180-49-68=63°.

Co-interior angles add to 180 degrees; they do not have to be equal.

1. The two marked angles are co-interior.
2. Co-interior angles on parallel lines add to 180°, so x=180-124=56.
3. The learner used equality instead of the straight-angle total.

Calculate the third triangle angle, then compare the corresponding angles.

Part (a)

1. Angles in a triangle add to 180°.
2. x=180-42-64=74°.

Part (b)

1. The marked 74° angle and the angle x occupy corresponding positions.
2. Equal corresponding angles show that p and q are parallel.

Find the whole obtuse angle above the lower line before halving it.

1. The angle above and to the right at the lower crossing is 66°, because corresponding angles on parallel lines are equal.
2. The whole angle above and to the left is 180-66=114°, because angles on a straight line add to 180°.
3. The bisector divides this angle equally, so a=114\div2=57°.

Share 180 degrees in the given ratio before finding the adjacent angle.

1. Co-interior angles on parallel lines add to 180°.
2. There are 2+3=5 equal parts, so each part is 180\div5=36°.
3. The smaller angle is 2\times36=72°.
4. Its adjacent angle on a straight line is 180-72=108°.

Set the two expressions equal. The question asks for an angle, not just x.

1. Corresponding angles on parallel lines are equal, so 3x+5=2x+30.
2. Subtract 2x and then 5: x=25.
3. Each angle is 3\times25+5=80°. Check: 2\times25+30=80°.

Find one base angle using the parallel lines. The two base angles are equal.

1. The left base angle is 52°, because alternate angles on parallel lines are equal.
2. The base angles are equal because the two ticked sides are equal, so both base angles are 52°.
3. Angles in a triangle add to 180°, so x=180-52-52=76°.