Challenge Problems | Math Coach Amy

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Key Ideas Angle Chase Parallel Lines What's Wrong

straight line: 180°

triangle: 180°

parallel: corresponding angles equal

What to Remember

Straight line and vertical angles

Angles that form a straight line add to 180°. Vertical angles are equal.

Example: a straight line shows 148°.

x + 148° = 180°

x = 180 − 148

x = 32°

Triangle sum

The three interior angles of a triangle add to 180°.

Example: two angles are 36° and 82°.

36° + 82° + y = 180°

36° + 82° = 118°

y = 180 − 118

y = 62°

Parallel lines

When two lines are parallel, corresponding angles are equal and alternate interior angles are equal. Consecutive interior angles add to 180°.

Example: a transversal cuts parallel lines, and one angle is 67°.

The corresponding angle is 67°.

z + 67° = 180°

z = 113°

A picture can be impossible

Arrow marks are a claim that lines are parallel. Angle measures have to agree with that claim.

If a pair of corresponding angles is not equal, the lines are not parallel.

If consecutive interior angles do not add to 180°, the lines are not parallel.

Challenge

Angle Chase

Vertical angles, a straight line, and the angles of a triangle.

Find the measure of each lettered angle. The 64° angle is inside the triangle at the left. The 47° angle is vertical to angle d.

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Vertical angles are equal.

d = 47°

The three angles of a triangle add to 180°.

64° + d + a = 180°

64° + 47° + a = 180°

64° + 47° = 111°

a = 180 − 111

a = 69°

Angles a and b form a straight line.

a + b = 180°

69° + b = 180°

b = 111°

Vertical angles are equal.

c = 64°

Angles d and e form a straight line.

d + e = 180°

47° + e = 180°

e = 133°

Answer: a = 69°; b = 111°; c = 64°; d = 47°; e = 133°

Find the measure of each lettered angle. The square marks a right angle. The 155° angle and angle b form a straight line. The 68° angle is inside the upper triangle.

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The square marks a right angle.

a = 90°

Angles b and 155° form a straight line.

b + 155° = 180°

b = 180 − 155

b = 25°

The lower triangle adds to 180°.

90° + 25° + c = 180°

c = 65°

Vertical angles are equal, so the upper triangle has a 25° angle at that same crossing.

The upper triangle adds to 180°.

25° + 68° + d = 180°

25° + 68° = 93°

d = 180 − 93

d = 87°

The 68° angle and angle e form a straight line.

68° + e = 180°

e = 112°

Answer: a = 90°; b = 25°; c = 65°; d = 87°; e = 112°

Challenge

Parallel Lines

Arrow marks show the parallel lines. A square marks a right angle.

Lines m and n are parallel. The arcs mark congruent angles. The 74° angle is the lower-left interior angle of the triangle. Find z.

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m ∥ n, and the left side of the triangle is a transversal.

The marked angle above m corresponds to the 74° angle.

That marked angle is 74°.

The arcs match, so the lower-right base angle is 74°.

The right side is a transversal. Angle z and the lower-right base angle are consecutive interior angles.

z + 74° = 180°

z = 180 − 74

z = 106°

Answer: z = 106°

Lines m and n are parallel. Find the measure of each lettered angle. The 112° angle is next to angle a, outside the top triangle. The 41° angle matches angle e by corresponding angles.

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Angles a and 112° form a straight line.

a + 112° = 180°

a = 68°

Vertical angles are equal.

b = 112°

m ∥ n. Corresponding angles are equal.

c = a = 68°

d = 112°

Corresponding angles are equal.

e = 41°

The top triangle adds to 180°.

a + e + f = 180°

68° + 41° + f = 180°

68° + 41° = 109°

f = 180 − 109

f = 71°

The 41° angle and angle g form a straight line.

41° + g = 180°

g = 139°

Answer: a = 68°; b = 112°; c = 68°; d = 112°; e = 41°; f = 71°; g = 139°

The arrow marks show parallel lines. The square marks a right angle. Find the measure of each lettered angle.

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Angles a and 134° form a straight line.

a + 134° = 180°

a = 46°

Corresponding angles are equal, and b is that angle in the right triangle.

b = 46°

The square marks a right angle, so that angle is 90°.

The right triangle adds to 180°.

46° + 90° + c = 180°

c = 44°

Angles d and 98° form a straight line.

d + 98° = 180°

d = 82°

Corresponding angles are equal.

e = 82°

Vertical angles are equal.

f = 98°

Answer: a = 46°; b = 46°; c = 44°; d = 82°; e = 82°; f = 98°

Matching arrow marks show which lines are parallel. The square marks a right angle. Find the measure of each lettered angle.

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Angles a and 125° form a straight line.

a + 125° = 180°

a = 55°

Corresponding angles on the double-arrow lines are equal.

b = 55°

The left triangle adds to 180°.

55° + 28° + c = 180°

55° + 28° = 83°

c = 180 − 83

c = 97°

Angles d and 116° form a straight line.

d + 116° = 180°

d = 64°

Corresponding angles on the single-arrow lines are equal.

e = 64°

The square marks a right angle, so that angle is 90°.

The right triangle adds to 180°.

64° + 90° + f = 180°

f = 26°

Vertical angles are equal.

g = 116°

Answer: a = 55°; b = 55°; c = 97°; d = 64°; e = 64°; f = 26°; g = 116°

Lines m and n are parallel. The square marks a right angle. The ray with the matching arcs bisects angle b. The 104° angle is next to angle a, and the 122° angle is next to angle e. Find the measure of each lettered angle.

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Angles a and 104° form a straight line.

a + 104° = 180°

a = 76°

Corresponding angles are equal.

b = 76°

The ray bisects angle b.

c = d = 76°2

c = 38°

d = 38°

Angles e and 122° form a straight line.

e + 122° = 180°

e = 58°

The triangle that uses c and e adds to 180°.

38° + 58° + f = 180°

38° + 58° = 96°

f = 180 − 96

f = 84°

The square marks a right angle, so that angle is 90°.

The right triangle that uses a adds to 180°.

90° + 76° + g = 180°

g = 14°

Answer: a = 76°; b = 76°; c = 38°; d = 38°; e = 58°; f = 84°; g = 14°

The arrow marks show parallel lines. The square marks a right angle. The 136° angle and angle a form a straight line. The 70° angle is inside the upper triangle. Find the measure of each lettered angle.

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Angles a and 136° form a straight line.

a + 136° = 180°

a = 44°

Corresponding angles are equal, so the lower triangle has a 44° angle.

b = 44°

The square marks a right angle, so that angle is 90°.

The lower triangle adds to 180°.

44° + 90° + c = 180°

c = 46°

Vertical angles are equal, so the upper triangle has a 44° angle.

The upper triangle adds to 180°.

44° + 70° + d = 180°

44° + 70° = 114°

d = 180 − 114

d = 66°

The 70° angle and angle e form a straight line.

70° + e = 180°

e = 110°

Corresponding angles are equal.

f = 136°

Answer: a = 44°; b = 44°; c = 46°; d = 66°; e = 110°; f = 136°

Challenge

What's Wrong

The measures and the arrow marks cannot all be true.

What is wrong with each picture? The arrow marks are a claim about which lines are parallel.

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(a) The single arrows say the two lines are parallel.

The 48° angle and the 121° angle are consecutive interior angles.

Consecutive interior angles add to 180° when the lines are parallel.

48° + 121° = 169°

169° is not 180°, so the lines cannot be parallel.

(b) The double arrows say the top and middle lines are parallel.

The 35° angle and the 50° angle are corresponding angles.

Corresponding angles are equal when the lines are parallel.

35° is not 50°, so those two lines cannot be parallel.

Answer: (a) 48° + 121° = 169°, not 180°, so the lines are not parallel. (b) Corresponding angles 35° and 50° are not equal, so the double-arrow lines are not parallel.

What is wrong with this picture?

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The arrows say the two lines are parallel.

The 142° angle and the 28° angle are corresponding angles.

Corresponding angles are equal when the lines are parallel.

142° is not 28°, so the lines cannot be parallel.

The 118° angle and the 62° angle form a straight line.

118° + 62° = 180°

That pair agrees with a straight line. The parallel marks do not agree with 142° and 28°.

Answer: The lines are marked parallel, but corresponding angles 142° and 28° are not equal.

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