Question

directions - complete the following problems, show all work. 1. use the same diagram for problems a and b. information does not carry over from problem to problem. a) m∠jkl = 46, m∠lkm = 18, m∠jkm = ____ b) m∠jkl = __, m∠lkm = 21, m∠jkm = 88 2. xb is the angle bisector of ∠axc, m∠axb = 23. find the following: m∠bxc = __, m∠axc = __ 3. m∠abc = 122, m∠abd = 8x + 20, m∠dbc = 22x - 3. find the following: x = __, m∠abd = __, m∠dbc = __ 4. m∠prs = 180, m∠prq = 6x + 13, m∠qrs = 3x - 4. find the following: x = __, m∠prq = __, m∠qrs = ____

Explanation:

Step1: Sum adjacent angles

$m\angle JKM = m\angle JKL + m\angle LKM = 46 + 18 = 64$

Step2: Subtract to find angle

$m\angle JKL = m\angle JKM - m\angle LKM = 88 - 21 = 67$

Step3: Angle bisector property

$m\angle BXC = m\angle AXB = 23$

Step4: Sum bisected angles

$m\angle AXC = m\angle AXB + m\angle BXC = 23 + 23 = 46$

Step5: Set up angle sum equation

$(8x + 20) + (22x - 3) = 122$

Step6: Solve for x

$30x + 17 = 122 \implies 30x = 105 \implies x = 3.5$

Step7: Calculate ∠ABD

$m\angle ABD = 8(3.5) + 20 = 48$

Step8: Calculate ∠DBC

$m\angle DBC = 22(3.5) - 3 = 74$

Step9: Set up straight angle equation

$(6x + 13) + (3x - 4) = 180$

Step10: Solve for x

$9x + 9 = 180 \implies 9x = 171 \implies x = 19$

Step11: Calculate ∠PRQ

$m\angle PRQ = 6(19) + 13 = 127$

Step12: Calculate ∠QRS

$m\angle QRS = 3(19) - 4 = 53$

Answer:

1A) 64
1B) 67

1. m∠BXC = 23, m∠AXC = 46
2. x = 3.5, m∠ABD = 48, m∠DBC = 74
3. x = 19, m∠PRQ = 127, m∠QRS = 53