Angle Addition Postulate | Math Coach Amy

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

m∠ABD + m∠DBC = m∠ABC

What to Remember

Angle Addition Postulate (numbers only)

If ray BD is between sides BA and BC, then m∠ABD + m∠DBC = m∠ABC.

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m∠ABD + m∠DBC = m∠ABC

28° + 17° = m∠ABC

m∠ABC = 45°

Angle Addition Postulate (variable angle measures)

Label the two pieces and the whole angle. Solve for x, then substitute.

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m∠ABD = 3x + 1, m∠DBC = 2x, and m∠ABC = 41°.

m∠ABD + m∠DBC = m∠ABC

(3x + 1) + 2x = 41

5x + 1 = 41

5x = 40

x = 8

m∠ABD = 3(8) + 1

m∠ABD = 25°

Easy

Find Total Measure

Add the two pieces. The first three diagrams are given. Draw the last three yourself.

Example

Ray BD is between sides BA and BC of ∠ABC. m∠ABD = 28° and m∠DBC = 17°. Find m∠ABC.

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m∠ABD + m∠DBC = m∠ABC

28° + 17° = m∠ABC

m∠ABC = 45°

Answer: 45°

Ray QS is between sides QP and QR of ∠PQR. m∠PQS = 22° and m∠SQR = 35°. Find m∠PQR.

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m∠PQS + m∠SQR = m∠PQR

22° + 35° = m∠PQR

m∠PQR = 57°

Answer: 57°

Ray BD is between sides BA and BC of ∠ABC. m∠ABD = 40° and m∠DBC = 40°. Find m∠ABC.

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Show Answer

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m∠ABD + m∠DBC = m∠ABC

40° + 40° = m∠ABC

m∠ABC = 80°

Answer: 80°

Draw the angle. Ray XZ is between sides XW and XY of ∠WXY. m∠WXZ = 15° and m∠ZXY = 48°. Find m∠WXY.

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m∠WXZ + m∠ZXY = m∠WXY

15° + 48° = m∠WXY

m∠WXY = 63°

Answer: 63°

Draw the angle. Ray FH is between sides FE and FG of ∠EFG. m∠EFH = 33° and m∠HFG = 21°. Find m∠EFG.

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m∠EFH + m∠HFG = m∠EFG

33° + 21° = m∠EFG

m∠EFG = 54°

Answer: 54°

Draw the angle. Ray NL is between sides NM and NP of ∠MNP. m∠MNL = 19° and m∠LNP = 19°. Find m∠MNP.

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m∠MNL + m∠LNP = m∠MNP

19° + 19° = m∠MNP

m∠MNP = 38°

Answer: 38°

Easy

Find Angle

One piece is missing. The first three diagrams are given. Draw the last three yourself.

Example

Ray BD is between sides BA and BC of ∠ABC. m∠ABC = 70° and m∠ABD = 25°. Find m∠DBC.

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m∠ABD + m∠DBC = m∠ABC

25° + m∠DBC = 70°

m∠DBC = 45°

Answer: 45°

Ray QS is between sides QP and QR of ∠PQR. m∠PQR = 90° and m∠PQS = 38°. Find m∠SQR.

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m∠PQS + m∠SQR = m∠PQR

38° + m∠SQR = 90°

m∠SQR = 52°

Answer: 52°

Ray XZ is between sides XW and XY of ∠WXY. m∠WXY = 55° and m∠WXZ = 12°. Find m∠ZXY.

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m∠WXZ + m∠ZXY = m∠WXY

12° + m∠ZXY = 55°

m∠ZXY = 43°

Answer: 43°

Draw the angle. Ray FH is between sides FE and FG of ∠EFG. m∠EFG = 100° and m∠EFH = 47°. Find m∠HFG.

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m∠EFH + m∠HFG = m∠EFG

47° + m∠HFG = 100°

m∠HFG = 53°

Answer: 53°

Draw the angle. Ray NQ is between sides NM and NP of ∠MNP. m∠MNP = 64° and m∠MNQ = 29°. Find m∠QNP.

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m∠MNQ + m∠QNP = m∠MNP

29° + m∠QNP = 64°

m∠QNP = 35°

Answer: 35°

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABC = 81° and m∠DBC = 36°. Find m∠ABD.

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m∠ABD + m∠DBC = m∠ABC

m∠ABD + 36° = 81°

m∠ABD = 45°

Answer: 45°

Medium

Mixed

Draw the angle. Solve for x, then find the measure.

Example

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABD = 2x + 5, m∠DBC = x + 7, and m∠ABC = 51°. Find m∠ABD.

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m∠ABD + m∠DBC = m∠ABC

(2x + 5) + (x + 7) = 51

3x + 12 = 51

3x = 39

x = 13

m∠ABD = 2(13) + 5

m∠ABD = 31°

Answer: 31°

Draw the angle. Ray QS is between sides QP and QR of ∠PQR. m∠PQS = 4x − 3, m∠SQR = 2x + 9, and m∠PQR = 60°. Find m∠PQS.

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m∠PQS + m∠SQR = m∠PQR

(4x − 3) + (2x + 9) = 60

6x + 6 = 60

6x = 54

x = 9

m∠PQS = 4(9) − 3

m∠PQS = 33°

Answer: 33°

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABD = 3x + 2, m∠DBC = x + 10, and m∠ABC = 5x. Find m∠ABC.

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m∠ABD + m∠DBC = m∠ABC

(3x + 2) + (x + 10) = 5x

4x + 12 = 5x

12 = x

m∠ABC = 5(12)

m∠ABC = 60°

Answer: 60°

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABD = x + 15, m∠DBC = 2x, and m∠ABC = 4x − 3. Find m∠ABC.

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m∠ABD + m∠DBC = m∠ABC

(x + 15) + 2x = 4x − 3

3x + 15 = 4x − 3

18 = x

m∠ABC = 4(18) − 3

m∠ABC = 69°

Answer: 69°

Draw the angle. Ray XZ is between sides XW and XY of ∠WXY. m∠WXZ = 5x − 4, m∠ZXY = x + 14, and m∠WXY = 4x + 20. Find m∠WXY.

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m∠WXZ + m∠ZXY = m∠WXY

(5x − 4) + (x + 14) = 4x + 20

6x + 10 = 4x + 20

2x = 10

x = 5

m∠WXY = 4(5) + 20

m∠WXY = 40°

Answer: 40°

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABD = 4x − 1, m∠DBC = 2x + 13, and m∠ABC = 5x + 16. Find m∠ABC.

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m∠ABD + m∠DBC = m∠ABC

(4x − 1) + (2x + 13) = 5x + 16

6x + 12 = 5x + 16

x = 4

m∠ABC = 5(4) + 16

m∠ABC = 36°

Answer: 36°

Hard

With Quadratics

Draw the angle. Set the equation equal to 0, then factor.

Example

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABD = x², m∠DBC = 5x, and m∠ABC = 36°. Find m∠ABD.

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m∠ABD + m∠DBC = m∠ABC

x² + 5x = 36

Set the equation equal to 0.

x² + 5x − 36 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −36 and b = 5. Those numbers are 9 and −4.

(x + 9)(x − 4) = 0

x = −9 or x = 4

x = −9 makes m∠DBC = 5(−9) = −45. An angle measure cannot be negative, so reject x = −9.

x = 4

m∠ABD = 4²

m∠ABD = 16°

Answer: 16°

Draw the angle. Ray QS is between sides QP and QR of ∠PQR. m∠PQS = x², m∠SQR = 2x, and m∠PQR = 35°. Find m∠PQS.

Show Answer

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m∠PQS + m∠SQR = m∠PQR

x² + 2x = 35

Set the equation equal to 0.

x² + 2x − 35 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −35 and b = 2. Those numbers are 7 and −5.

(x + 7)(x − 5) = 0

x = −7 or x = 5

x = −7 makes m∠SQR = 2(−7) = −14. An angle measure cannot be negative, so reject x = −7.

x = 5

m∠PQS = 5²

m∠PQS = 25°

Answer: 25°

Draw the angle. Ray BD is between sides BA and BC of ∠ABC. m∠ABD = x², m∠DBC = 4x, and m∠ABC = 45°. Find m∠ABD.

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m∠ABD + m∠DBC = m∠ABC

x² + 4x = 45

Set the equation equal to 0.

x² + 4x − 45 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −45 and b = 4. Those numbers are 9 and −5.

(x + 9)(x − 5) = 0

x = −9 or x = 5

x = −9 makes m∠DBC = 4(−9) = −36. An angle measure cannot be negative, so reject x = −9.

x = 5

m∠ABD = 5²

m∠ABD = 25°

Answer: 25°

Draw the angle. Ray XZ is between sides XW and XY of ∠WXY. m∠WXZ = x², m∠ZXY = 3x, and m∠WXY = 40°. Find m∠WXZ.

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m∠WXZ + m∠ZXY = m∠WXY

x² + 3x = 40

Set the equation equal to 0.

x² + 3x − 40 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −40 and b = 3. Those numbers are 8 and −5.

(x + 8)(x − 5) = 0

x = −8 or x = 5

x = −8 makes m∠ZXY = 3(−8) = −24. An angle measure cannot be negative, so reject x = −8.

x = 5

m∠WXZ = 5²

m∠WXZ = 25°

Answer: 25°

Draw the angle. Ray FH is between sides FE and FG of ∠EFG. m∠EFH = x², m∠HFG = 4x, and m∠EFG = 60°. Find m∠EFH.

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m∠EFH + m∠HFG = m∠EFG

x² + 4x = 60

Set the equation equal to 0.

x² + 4x − 60 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −60 and b = 4. Those numbers are 10 and −6.

(x + 10)(x − 6) = 0

x = −10 or x = 6

x = −10 makes m∠HFG = 4(−10) = −40. An angle measure cannot be negative, so reject x = −10.

x = 6

m∠EFH = 6²

m∠EFH = 36°

Answer: 36°

Draw the angle. Ray NL is between sides NM and NP of ∠MNP. m∠MNL = x², m∠LNP = 2x, and m∠MNP = 48°. Find m∠MNL.

Show Answer

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m∠MNL + m∠LNP = m∠MNP

x² + 2x = 48

Set the equation equal to 0.

x² + 2x − 48 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −48 and b = 2. Those numbers are 8 and −6.

(x + 8)(x − 6) = 0

x = −8 or x = 6

x = −8 makes m∠LNP = 2(−8) = −16. An angle measure cannot be negative, so reject x = −8.

x = 6

m∠MNL = 6²

m∠MNL = 36°

Answer: 36°

Hard

Three Angles

Draw the angle. Three pieces share the same vertex.

Example

Draw the angle. Rays BD and BE are between sides BA and BC of ∠ABC, with D between A and E. m∠ABD = 20°, m∠DBE = 35°, and m∠EBC = 15°. Find m∠ABC and m∠ABE.

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m∠ABE = 20° + 35°

m∠ABE = 55°

m∠ABC = 20° + 35° + 15°

m∠ABC = 70°

Answer: m∠ABC = 70°, m∠ABE = 55°

Draw the angle. Rays QS and QR are between sides QP and QT of ∠PQT, with S between P and R. m∠PQS = 12°, m∠SQR = 28°, and m∠RQT = 17°. Find m∠PQT and m∠PQR.

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m∠PQR = 12° + 28°

m∠PQR = 40°

m∠PQT = 12° + 28° + 17°

m∠PQT = 57°

Answer: m∠PQT = 57°, m∠PQR = 40°

Draw the angle. Rays BD and BE are between sides BA and BC of ∠ABC, with D between A and E. m∠ABD = 30°, m∠DBE = 22°, and m∠EBC = 18°. Find m∠ABC and m∠DBC.

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m∠DBC = 22° + 18°

m∠DBC = 40°

m∠ABC = 30° + 22° + 18°

m∠ABC = 70°

Answer: m∠ABC = 70°, m∠DBC = 40°

Draw the angle. Rays BD and BE are between sides BA and BC of ∠ABC, with D between A and E. m∠ABD = 2x + 1, m∠DBE = x + 4, m∠EBC = 11°, and m∠ABC = 4x. Find m∠ABC.

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(2x + 1) + (x + 4) + 11 = 4x

3x + 16 = 4x

16 = x

m∠ABC = 4(16)

m∠ABC = 64°

Answer: 64°

Draw the angle. Rays QS and QR are between sides QP and QT of ∠PQT, with S between P and R. m∠PQS = 3x − 2, m∠SQR = x + 5, m∠RQT = 6°, and m∠PQT = 5x + 1. Find m∠PQT.

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(3x − 2) + (x + 5) + 6 = 5x + 1

4x + 9 = 5x + 1

8 = x

m∠PQT = 5(8) + 1

m∠PQT = 41°

Answer: 41°

Draw the angle. Rays BD and BE are between sides BA and BC of ∠ABC, with D between A and E. m∠ABD = x + 3, m∠DBE = 2x + 1, m∠EBC = x − 2, and m∠ABC = 5x − 6. Find m∠ABC.

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(x + 3) + (2x + 1) + (x − 2) = 5x − 6

4x + 2 = 5x − 6

8 = x

m∠ABC = 5(8) − 6

m∠ABC = 34°

Answer: 34°

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