Question

example 2

1. rays ba and bc are perpendicular. point d lies in the interior of ∠abc. if m∠abd = (3r + 5)° and m∠dbc = (5r − 27)°, find m∠abd and m∠dbc.
2. $overleftrightarrow{wx}$ and $overleftrightarrow{yz}$ intersect at point v. if m∠wvy = (4a + 58)° and m∠xvy = (2b − 18)°, find the values of a and b such that $overleftrightarrow{wx}$ is perpendicular to $overleftrightarrow{yz}$.
3. refer to the figure at the right. if m∠2 = (a + 15)° and m∠3 = (a + 35)°, find the value of a such that $overleftrightarrow{hl} perp overleftrightarrow{hj}$.
4. rays da and dc are perpendicular. point b lies in the interior of ∠adc. if m∠adb = (3a + 10)° and m∠bdc = 13a°, find a, m∠adb, and m∠bdc.

diagram: g---h---i (horizontal line), with hj, hk, hl, hi; angles 1,2,3,4, with angle 3 and 4 as right angle?

Explanation:

Problem 7

Step1: Since BA ⊥ BC, ∠ABC = 90°. So ∠ABD + ∠DBC = 90°.

$(3r + 5) + (5r - 27) = 90$

Step2: Simplify the equation.

$8r - 22 = 90$

Step3: Solve for r.

$8r = 112$ → $r = 14$

Step4: Find m∠ABD.

$3(14) + 5 = 47$

Step5: Find m∠DBC.

$5(14) - 27 = 43$

Step1: If WX ⊥ YZ, then ∠WVY = 90° and ∠XVY = 90°.

For ∠WVY: $4a + 58 = 90$
For ∠XVY: $2b - 18 = 90$

Step2: Solve for a.

$4a = 32$ → $a = 8$

Step3: Solve for b.

$2b = 108$ → $b = 54$

Step1: Since HL ⊥ HJ, ∠2 + ∠3 = 90°.

$(a + 15) + (a + 35) = 90$

Step2: Simplify the equation.

$2a + 50 = 90$

Step3: Solve for a.

$2a = 40$ → $a = 20$

Answer:

m∠ABD = 47°, m∠DBC = 43°

Problem 8