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Math Coach Amy
Amy Ferguson Moncure

Geometry · Unit 3

Angle Chase

Combine triangle sum, linear pairs, isosceles base angles, and parallel-line relationships to find every missing angle.

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Key Ideas Easy Medium Hard Missing Angles

triangle: 180°

straight line: 180°

parallel: corresponding angles equal

What to Remember

Triangle sum

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

Example: 40° and 65° in a triangle leave 75° for the third angle.

Straight line

Angles on a straight line add to 180°. If a triangle sits on a line, the interior base angle and the exterior next to it are a linear pair.

Example: interior 72° means the adjacent exterior is 108°.

Isosceles

If two sides are congruent, the base angles are congruent.

Example: AE ≅ AB and the vertex angle is 32°, so each base angle is 180 − 322 = 74°.

Parallel plus a triangle

Mark corresponding or alternate interior angles first, then finish the triangle.

Two lines perpendicular to the same line are parallel.

Several missing angles

On parallel lines, corresponding angles match.

Vertical angles match. A linear pair adds to 180°.

Find one letter, then use it to find the next.

Level 1

Easy

One triangle, maybe a linear pair on a line.

Example
Triangle ABC sits on a straight line at C. m∠A = 42° and m∠B = 66°. Find m∠ACB and the adjacent exterior angle at C.

Triangle sum: 42 + 66 + m∠ACB = 180

m∠ACB = 72°

Linear pair: exterior = 180 − 72 = 108°

Answer: m∠ACB = 72°; exterior 108°

1
In triangle PQR, m∠P = 50° and m∠Q = 60°. Find m∠R.
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50 + 60 + m∠R = 180

m∠R = 70°

Answer: 70°

2
A triangle on a line has interior base angle 55°. Find the adjacent exterior angle on the line.
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Linear pair: 180 − 55 = 125°

Answer: 125°

3
Triangle DEF is isosceles with DE ≅ DF and vertex m∠D = 40°. Find a base angle.
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180 − 402 = 1402 = 70

Each base angle is 70°.

Answer: 70°

4
Vertical angles at a point on a line measure 35°. The adjacent angle on the line is?
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Linear pair with 35° is 145°.

Answer: 145°

5
In triangle ABC, m∠A = 90° and m∠B = 35°. Find m∠C.
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90 + 35 + m∠C = 180

m∠C = 55°

Answer: 55°

Level 2

Medium

Two triangles, or a line through a triangle.

Example
In triangle ABC, D is on AC. m∠BAC = 40°, m∠ABD = 25°, and m∠DBC = 25°. Find m∠ADB, m∠BDC, and m∠BCD.

Triangle ABD: 40 + 25 + m∠ADB = 180

m∠ADB = 115°

Linear pair at D on AC: m∠BDC = 180 − 115 = 65°

Triangle BDC: 65 + 25 + m∠BCD = 180

m∠BCD = 90°

Answer: m∠ADB = 115°; m∠BDC = 65°; m∠BCD = 90°

1
Two triangles sit on a straight line and share vertex C. In the left triangle, m∠A = 42° and m∠B = 66°. In the right triangle, the right base m∠E = 40°. Find the left base of the right triangle and the vertex of the right triangle.
Show Answer

Left triangle at C: 42 + 66 + m∠ACB = 180, so m∠ACB = 72°.

Linear pair at C: left base of the right triangle = 180 − 72 = 108°.

Right triangle: 108 + 40 + vertex = 180, so the vertex is 32°.

Answer: 108° and 32°

2
Triangle ABC is isosceles with AB ≅ AC. m∠B = 58°. Find m∠A.
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Base angles are equal, so m∠C = 58°.

m∠A = 180 − 58 − 58 = 64°

Answer: 64°

3
A line is parallel to side BC of triangle ABC. The corresponding angle to m∠A is 47°, and m∠B = 62°. Find m∠C.
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Corresponding angles: m∠A = 47°.

47 + 62 + m∠C = 180

m∠C = 71°

Answer: 71°

4
On a straight line, a triangle has interior angles 38° (left base) and 70° (vertex). Find the right-base interior angle and the exterior angle to its right.
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Interior right base: 180 − 38 − 70 = 72°

Exterior: 180 − 72 = 108°

Answer: 72° and 108°

5
In triangle ABC, D is on AC. m∠A = 4x + 8, m∠ABD = 3x + 1, and m∠DBC = 2x + 7. Given that x = 8, find m∠ADB, m∠BDC, and m∠BCD.
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m∠A = 4(8) + 8 = 40°

m∠ABD = 3(8) + 1 = 25°

m∠DBC = 2(8) + 7 = 23°

Triangle ABD: 40 + 25 + m∠ADB = 180

m∠ADB = 115°

Linear pair at D: m∠BDC = 180 − 115 = 65°

Triangle BDC: 65 + 23 + m∠BCD = 180

m∠BCD = 92°

Answer: m∠ADB = 115°; m∠BDC = 65°; m∠BCD = 92°

Level 3

Hard

Solve for x in the diagram, then chase every requested angle. Isosceles plus perpendiculars.

Example
In triangle ABC, D is on AC. m∠A = 5x + 5, m∠ABD = 4x − 1, and m∠DBC = 2x + 9. Use triangle ABC together with the fact that m∠C = 2x + 11 to find x, then find m∠ADB, m∠BDC, and m∠BCD.

m∠ABC = (4x − 1) + (2x + 9) = 6x + 8

Triangle ABC: (5x + 5) + (6x + 8) + (2x + 11) = 180

13x + 24 = 180

13x = 156

x = 12

m∠A = 5(12) + 5 = 65°

m∠ABD = 4(12) − 1 = 47°

m∠DBC = 2(12) + 9 = 33°

m∠C = 2(12) + 11 = 35°

Triangle ABD: 65 + 47 + m∠ADB = 180, so m∠ADB = 68°

m∠BDC = 180 − 68 = 112°

Check triangle BDC: 112 + 33 + 35 = 180.

Answer: x = 12; m∠ADB = 68°; m∠BDC = 112°; m∠BCD = 35°

1
In the diagram, AE ⊥ AF and CF ⊥ AF, so AE ∥ CF. Also AE ≅ AB, m∠EAB = 32°. Find m∠ABE.
Show Answer

AE ⊥ AF, so m∠EAF = 90°.

m∠BAF = 90 − 32 = 58°.

Triangle AEB is isosceles with AE ≅ AB and vertex m∠EAB = 32°.

Base angles: 180 − 322 = 74°.

m∠ABE = 74°

Answer: 74°

2
AE ⊥ AF, CF ⊥ AF, and AE ≅ AB, with m∠EAB = 32°. Points E, B, and C are collinear. Find m∠ABC.
Show Answer

m∠ABE = 74° from the isosceles triangle.

E–B–C is a straight line, so m∠ABE and m∠ABC are a linear pair.

m∠ABC = 180 − 74 = 106°

Answer: 106°

3
Two triangles sit on a line as in the diagram. m∠1 = 38° and m∠7 = 104°. The left legs of the two triangles are parallel, and the right legs are parallel. Find m∠2, m∠3, m∠6, and m∠8.
Show Answer

The left legs are parallel, so corresponding base angles: m∠3 = m∠1 = 38°.

Right triangle: 38 + 104 + m∠6 = 180

m∠6 = 38°

The right legs are parallel, so the left triangle matches: m∠2 = m∠6 = 38°.

Linear pair of ∠7: m∠8 = 180 − 104 = 76°.

Answer: m∠2 = 38°; m∠3 = 38°; m∠6 = 38°; m∠8 = 76°

4
In triangle ABC, D is on AC. m∠A = 3x + 10, m∠ABD = 2x + 5, m∠DBC = x + 15, and m∠C = 4x − 10. Find x and m∠ADB.
Show Answer

m∠ABC = (2x + 5) + (x + 15) = 3x + 20

Triangle ABC: (3x + 10) + (3x + 20) + (4x − 10) = 180

10x + 20 = 180

10x = 160

x = 16

m∠A = 58, m∠ABD = 37

Triangle ABD: 58 + 37 + m∠ADB = 180

m∠ADB = 85°

Answer: x = 16; m∠ADB = 85°

5
Triangle PQR has m∠P = 28° and m∠Q = 84°. A line through R is parallel to PQ. Find the two angles the parallel line makes with PR and QR that are consecutive interior with ∠P and ∠Q.
Show Answer

m∠R = 180 − 28 − 84 = 68°

Consecutive interior with ∠P is 180 − 28 = 152°.

Consecutive interior with ∠Q is 180 − 84 = 96°.

Check: 152 + 96 + 68 would overcount; those two live on the parallel line at R’s sides, not inside the triangle.

Corresponding to ∠P on the parallel is 28°; consecutive interior is 152°.

Answer: 152° and 96°

Lines

Find Each Missing Angle

Arrow marks mean the lines are parallel. Find a, b, and c.

Example
The arrow marks show parallel lines. Find a, b, and c.

Not drawn to scale.

a is vertical to 74°.

a = 74°

b is a linear pair with 74°.

b = 180° − 74°

b = 106°

c is corresponding to 74°.

c = 74°

Answer: a = 74°; b = 106°; c = 74°

1
The arrow marks show parallel lines. Find a, b, and c.

Not drawn to scale.

Show Answer

a is corresponding to 63°.

a = 63°

b is corresponding to 63°.

b = 63°

c is a linear pair with b.

c = 180° − 63°

c = 117°

Answer: a = 63°; b = 63°; c = 117°

2
The arrow marks show parallel lines. Find a, b, and c.

Not drawn to scale.

Show Answer

a is vertical to 118°.

a = 118°

b is corresponding to 118°.

b = 118°

c is a linear pair with b.

c = 180° − 118°

c = 62°

Answer: a = 118°; b = 118°; c = 62°

3
The arrow marks show parallel lines. Find a, b, and c.

Not drawn to scale.

Show Answer

a is corresponding to 55°.

a = 55°

b is corresponding to 80°.

b = 80°

c is a linear pair with 80°.

c = 180° − 80°

c = 100°

Answer: a = 55°; b = 80°; c = 100°

4
The arrow marks show parallel lines. Find a, b, and c.

Not drawn to scale.

Show Answer

a is vertical to 152°.

a = 152°

b is corresponding to a.

b = 152°

c is a linear pair with a.

c = 180° − 152°

c = 28°

Answer: a = 152°; b = 152°; c = 28°

5
The arrow marks show parallel lines. Find a, b, and c.

Not drawn to scale.

Show Answer

a is corresponding to 46°.

a = 46°

The upper-left angle on that same transversal is 46°, so the upper-right angle is 180° − 46° = 134°.

b is that upper-right angle, moved down by corresponding angles.

b = 134°

c is corresponding to 105°.

c = 105°

Answer: a = 46°; b = 134°; c = 105°

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