Parallel lines and angle reasons

An angle rule needs a condition

Adjacent angles forming a straight angle total 180°. A complete turn around a point totals 360°. Vertically opposite angles at two intersecting straight lines are equal. “They look opposite” or “they touch” is not enough: identify the actual lines and regions.

An acute angle is between 0° and 90°, a right angle is 90°, an obtuse angle lies between 90° and 180°, and a reflex angle lies between 180° and 360°. A reflex angle and the smaller angle around the same rays total 360°.

A transversal cuts two parallel lines

Corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side of the transversal total 180° when the two lines are parallel. A useful first step is to mark the stated parallel pair and trace the transversal.

Here a = b by corresponding angles and a = c by alternate angles. If a = 58°, both equal 58°. The neighbouring angle on either horizontal line is 122°. If the lines were not known parallel, the crossing shape alone would not justify those equalities.

Check your understanding 1 Do not infer parallelism from the picture

A transversal cuts two lines stated to be parallel. An interior angle at one crossing is 64°. Find its corresponding angle and its same-side interior partner, stating a reason for each.

Your answer or drawing on paper

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The corresponding angle is 64° because the lines are parallel. The interior partner is 116° because the same-side interior pair totals 180°. Without the parallel condition, these results would not follow.

Work from a reflex angle

Two rays have a reflex angle of 248°. Find the smaller angle between them and classify it.