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Math  /  Geometry

Question9.03−9.069.03-9.06 Question 4 of 7 (1 point) I Question Attempt: 1 of Unlimited
In the figure below, m∠ABD=111∘,m∠CBD=64∘m \angle A B D=111^{\circ}, m \angle C B D=64^{\circ}, and BE‾\overline{B E} bisects ∠CBD\angle C B D. Find m∠ABEm \angle A B E. m∠ABE=□∘m \angle A B E=\square^{\circ}

Studdy Solution

STEP 1

1. Angles are measured in degrees.
2. BE‾\overline{BE} bisects ∠CBD\angle CBD, meaning ∠CBE=∠EBD\angle CBE = \angle EBD.
3. We need to find the measure of ∠ABE\angle ABE using the given angle measures.

STEP 2

1. Determine ∠CBD\angle CBD using the given angles.
2. Use the property of angle bisector to find ∠CBE\angle CBE and ∠EBD\angle EBD.
3. Find ∠ABE\angle ABE by summing ∠ABD\angle ABD and ∠EBD\angle EBD.

STEP 3

Calculate ∠CBD\angle CBD using the given angles ∠ABD\angle ABD and ∠CBD\angle CBD.
∠ABD+∠CBD=180∘ \angle ABD + \angle CBD = 180^\circ 111∘+∠CBD=180∘ 111^\circ + \angle CBD = 180^\circ ∠CBD=180∘−111∘ \angle CBD = 180^\circ - 111^\circ ∠CBD=69∘ \angle CBD = 69^\circ

STEP 4

Since BE‾\overline{BE} bisects ∠CBD\angle CBD, determine the measures of ∠CBE\angle CBE and ∠EBD\angle EBD.
∠CBE=∠EBD=∠CBD2 \angle CBE = \angle EBD = \frac{\angle CBD}{2} ∠CBE=∠EBD=69∘2 \angle CBE = \angle EBD = \frac{69^\circ}{2} ∠CBE=∠EBD=34.5∘ \angle CBE = \angle EBD = 34.5^\circ

STEP 5

Now, find ∠ABE\angle ABE by summing ∠ABD\angle ABD and ∠EBD\angle EBD.
∠ABE=∠ABD+∠EBD \angle ABE = \angle ABD + \angle EBD ∠ABE=111∘+34.5∘ \angle ABE = 111^\circ + 34.5^\circ ∠ABE=145.5∘ \angle ABE = 145.5^\circ
The solution is: m∠ABE=145.5∘ m \angle ABE = 145.5^\circ

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