Angle Bisectors | Math Coach Amy

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Key Ideas Find Total Measure Find Angle Mixed With Quadratics Three Angles

bisector: two congruent angles

m∠ABD = m∠DBC

What to Remember

Angle Bisector (numbers only)

An angle bisector splits an angle into two congruent angles.

If ray BD bisects ∠ABC and m∠ABD = 22°, then m∠DBC = 22° and m∠ABC = 44°.

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m∠ABD = m∠DBC

m∠ABD = m∠DBC = 22°

m∠ABC = 22° + 22°

m∠ABC = 44°

Angle Bisector (variable angle measures)

The two pieces are equal. Set them equal, solve, then plug back.

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m∠ABD = 3x + 4 and m∠DBC = 5x − 12.

m∠ABD = m∠DBC

3x + 4 = 5x − 12

16 = 2x

x = 8

m∠ABD = 3(8) + 4 = 28°

m∠ABC = 28° + 28° = 56°

Easy

Find Total Measure

Double the piece you know. Problems 1 and 2 include a diagram. It is not necessarily drawn to scale. Draw the last three yourself.

Example

Ray BD bisects ∠ABC. m∠ABD = 18°. Find m∠ABC.

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m∠ABD = m∠DBC

m∠ABD = m∠DBC = 18°

m∠ABC = 18° + 18°

m∠ABC = 36°

Answer: 36°

Ray QS bisects ∠PQR. m∠PQS = 25°. Find m∠PQR.

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m∠PQS = m∠SQR

m∠PQS = m∠SQR = 25°

m∠PQR = 25° + 25°

m∠PQR = 50°

Answer: 50°

Ray BD bisects ∠ABC. m∠ABD = 31°. Find m∠ABC.

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m∠ABD = m∠DBC

m∠ABD = m∠DBC = 31°

m∠ABC = 31° + 31°

m∠ABC = 62°

Answer: 62°

Draw the angle. Ray XZ bisects ∠WXY. m∠WXZ = 14°. Find m∠WXY.

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m∠WXZ = m∠ZXY

m∠WXZ = m∠ZXY = 14°

m∠WXY = 14° + 14°

m∠WXY = 28°

Answer: 28°

Draw the angle. Ray FH bisects ∠EFG. m∠EFH = 40°. Find m∠EFG.

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m∠EFH = m∠HFG

m∠EFH = m∠HFG = 40°

m∠EFG = 40° + 40°

m∠EFG = 80°

Answer: 80°

Draw the angle. Ray NL bisects ∠MNP. m∠MNL = 9°. Find m∠MNP.

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m∠MNL = m∠LNP

m∠MNL = m∠LNP = 9°

m∠MNP = 9° + 9°

m∠MNP = 18°

Answer: 18°

Easy

Find Angle

The whole angle is given. Find one congruent piece. Problems 1 and 2 include a diagram. It is not necessarily drawn to scale. Draw the last three yourself.

Example

m∠PQR = 80°. Ray QS bisects ∠PQR. Find m∠PQS.

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m∠PQS = m∠SQR

m∠PQS = 802

m∠PQS = 40°

Answer: 40°

m∠ABC = 54°. Ray BD bisects ∠ABC. Find m∠ABD.

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m∠ABD = m∠DBC

m∠ABD = 542

m∠ABD = 27°

Answer: 27°

m∠WXY = 90°. Ray XZ bisects ∠WXY. Find m∠WXZ.

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m∠WXZ = m∠ZXY

m∠WXZ = 902

m∠WXZ = 45°

Answer: 45°

Draw the angle. m∠EFG = 66°. Ray FH bisects ∠EFG. Find m∠EFH.

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m∠EFH = m∠HFG

m∠EFH = 662

m∠EFH = 33°

Answer: 33°

Draw the angle. m∠MNP = 120°. Ray NQ bisects ∠MNP. Find m∠MNQ.

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m∠MNQ = m∠QNP

m∠MNQ = 1202

m∠MNQ = 60°

Answer: 60°

Draw the angle. m∠JKL = 48°. Ray KM bisects ∠JKL. Find m∠JKM.

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m∠JKM = m∠MKL

m∠JKM = 482

m∠JKM = 24°

Answer: 24°

Medium

Mixed

Problems 1 and 2 include a diagram. It is not necessarily drawn to scale. Draw the last three yourself. Set the pieces equal, solve, then find the measure.

Example

Ray BD bisects ∠ABC. m∠ABD = 2x + 11 and m∠DBC = 4x − 5. Find m∠ABC.

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m∠ABD = m∠DBC

2x + 11 = 4x − 5

16 = 2x

x = 8

m∠ABD = 2(8) + 11 = 27°

m∠ABC = 27° + 27° = 54°

Answer: 54°

Ray BD bisects ∠ABC. m∠ABD = 5x − 3 and m∠DBC = 3x + 13. Find m∠ABC.

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5x − 3 = 3x + 13

2x = 16

x = 8

m∠ABD = 5(8) − 3 = 37°

m∠ABC = 74°

Answer: 74°

Ray QS bisects ∠PQR. m∠PQS = x + 20 and m∠SQR = 3x. Find m∠PQR.

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x + 20 = 3x

20 = 2x

x = 10

m∠PQS = 30°

m∠PQR = 60°

Answer: 60°

Draw the angle. Ray BD bisects ∠ABC. m∠ABD = 4x + 1 and m∠DBC = 2x + 17. Find m∠ABC.

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4x + 1 = 2x + 17

2x = 16

x = 8

m∠ABD = 33°

m∠ABC = 66°

Answer: 66°

Draw the angle. Ray BD bisects ∠ABC. m∠ABD = x + 12 and m∠ABC = 4x + 6. Find m∠ABC.

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The whole angle is twice one piece.

2(x + 12) = 4x + 6

2x + 24 = 4x + 6

18 = 2x

x = 9

m∠ABC = 4(9) + 6

m∠ABC = 42°

Answer: 42°

Draw the angle. Ray BD bisects ∠ABC. m∠ABD = 5x − 4 and m∠ABC = 6x + 10. Find m∠ABD.

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2(5x − 4) = 6x + 10

10x − 8 = 6x + 10

4x = 18

x = 92

m∠ABD = 592 − 4

m∠ABD = 452 − 82

m∠ABD = 372

Answer: 372°

Hard

With Quadratics

Problems 1 and 2 include a diagram. It is not necessarily drawn to scale. Draw the last three yourself. Set the equation equal to 0, then factor.

Example

Ray BD bisects ∠ABC. m∠ABD = x² − 10 and m∠DBC = 3x. Find m∠ABC.

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m∠ABD = m∠DBC

x² − 10 = 3x

Set the equation equal to 0.

x² − 3x − 10 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −10 and b = −3. Those numbers are −5 and 2.

(x − 5)(x + 2) = 0

x = 5 or x = −2

x = −2 makes m∠ABD = 4 − 10 = −6. An angle measure cannot be negative, so reject x = −2.

x = 5

m∠ABD = 25 − 10 = 15°

m∠ABC = 15° + 15° = 30°

Answer: 30°

Ray BD bisects ∠ABC. m∠ABD = x² − 12 and m∠DBC = 4x. Find m∠ABC.

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m∠ABD = m∠DBC

x² − 12 = 4x

Set the equation equal to 0.

x² − 4x − 12 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −12 and b = −4. Those numbers are −6 and 2.

(x − 6)(x + 2) = 0

x = 6 or x = −2

x = −2 makes m∠ABD = 4 − 12 = −8. An angle measure cannot be negative, so reject x = −2.

x = 6

m∠ABD = 36 − 12 = 24°

m∠ABC = 24° + 24° = 48°

Answer: 48°

Ray QS bisects ∠PQR. m∠PQS = x² − 18 and m∠SQR = 3x. Find m∠PQR.

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m∠PQS = m∠SQR

x² − 18 = 3x

Set the equation equal to 0.

x² − 3x − 18 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −18 and b = −3. Those numbers are −6 and 3.

(x − 6)(x + 3) = 0

x = 6 or x = −3

x = −3 makes m∠PQS = 9 − 18 = −9. An angle measure cannot be negative, so reject x = −3.

x = 6

m∠PQS = 36 − 18 = 18°

m∠PQR = 18° + 18° = 36°

Answer: 36°

Draw the angle. Ray XZ bisects ∠WXY. m∠WXZ = x² − 16 and m∠ZXY = 6x. Find m∠WXY.

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m∠WXZ = m∠ZXY

x² − 16 = 6x

Set the equation equal to 0.

x² − 6x − 16 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −16 and b = −6. Those numbers are −8 and 2.

(x − 8)(x + 2) = 0

x = 8 or x = −2

x = −2 makes m∠WXZ = 4 − 16 = −12. An angle measure cannot be negative, so reject x = −2.

x = 8

m∠WXZ = 64 − 16 = 48°

m∠WXY = 48° + 48° = 96°

Answer: 96°

Draw the angle. Ray FH bisects ∠EFG. m∠EFH = x² − 32 and m∠HFG = 4x. Find m∠EFG.

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m∠EFH = m∠HFG

x² − 32 = 4x

Set the equation equal to 0.

x² − 4x − 32 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −32 and b = −4. Those numbers are −8 and 4.

(x − 8)(x + 4) = 0

x = 8 or x = −4

x = −4 makes m∠EFH = 16 − 32 = −16. An angle measure cannot be negative, so reject x = −4.

x = 8

m∠EFH = 64 − 32 = 32°

m∠EFG = 32° + 32° = 64°

Answer: 64°

Draw the angle. Ray NL bisects ∠MNP. m∠MNL = x² − 21 and m∠LNP = 4x. Find m∠MNP.

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m∠MNL = m∠LNP

x² − 21 = 4x

Set the equation equal to 0.

x² − 4x − 21 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −21 and b = −4. Those numbers are −7 and 3.

(x − 7)(x + 3) = 0

x = 7 or x = −3

x = −3 makes m∠MNL = 9 − 21 = −12. An angle measure cannot be negative, so reject x = −3.

x = 7

m∠MNL = 49 − 21 = 28°

m∠MNP = 28° + 28° = 56°

Answer: 56°

Hard

Three Angles

Problems 1 and 2 include a diagram. It is not necessarily drawn to scale. Draw the last three yourself. One bisector can sit inside another.

Example

Ray AI bisects ∠KAN and ray AR bisects ∠KAI. m∠RAK = 15°. Find m∠RAN.

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AR bisects ∠KAI, so m∠RAI = 15° too.

m∠KAI = 15° + 15° = 30°

AI bisects ∠KAN, so m∠IAN = 30°.

m∠RAN = 15° + 30°

m∠RAN = 45°

Answer: 45°

Ray AI bisects ∠KAN and ray AR bisects ∠KAI. m∠RAK = 18°. Find m∠KAN.

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m∠KAI = 18° + 18° = 36°

AI bisects ∠KAN, so the whole angle is twice m∠KAI.

m∠KAN = 36° + 36°

m∠KAN = 72°

Answer: 72°

Ray AI bisects ∠KAN and ray AR bisects ∠KAI. m∠RAK = 10°. Find m∠RAN.

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m∠KAI = 20°

m∠IAN = 20°

m∠RAN = 10° + 20°

m∠RAN = 30°

Answer: 30°

Draw the angle. Ray AI bisects ∠KAN and ray AR bisects ∠KAI. m∠RAK = 22°. Find m∠KAN.

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m∠KAI = 44°

m∠KAN = 44° + 44°

m∠KAN = 88°

Answer: 88°

Draw the angle. Ray AI bisects ∠KAN and ray AR bisects ∠KAI. m∠RAK = 16°. Find m∠RAN.

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m∠KAI = 32°

m∠IAN = 32°

m∠RAN = 16° + 32°

m∠RAN = 48°

Answer: 48°

Draw the angle. Ray AI bisects ∠KAN and ray AR bisects ∠KAI. m∠RAK = x + 5 and m∠KAN = 6x + 2. Find m∠KAN.

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The whole angle is 4 times the smallest piece.

4(x + 5) = 6x + 2

4x + 20 = 6x + 2

18 = 2x

x = 9

m∠KAN = 6(9) + 2

m∠KAN = 56°

Answer: 56°

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