In the diagram below of triangle ABC, D is a midpoint of AB and E is a midpoint of BC. If mZDEB = 119 - 7x, and mZACE = 101 - 5x, what is the measure of ZACE? B D E A С

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31-10-2022
Mathematics

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In the diagram below of triangle ABC, D is a midpoint of AB and E is a midpoint of BC. If mZDEB = 119 - 7x, and mZACE = 101 - 5x, what is the measure of ZACE? B D E A С

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AlexiannaO605340 AlexiannaO605340 · Answer 1 · 2022-10-31T02:52:37-04:00

We want to find the measure of the ∠ACE. For doing so, we are going to use some geometrical properties with the information given.

As D is the midpoint of AB, we have that:

[tex]AD\cong DB[/tex]

And as E is the midpoint of BC, we have:

[tex]BE\cong EC[/tex]

Moreso, we can say that:

[tex]\frac{AB}{DB}=\frac{BC}{BE}=2[/tex]

And thus, the sides are proportional. As both triangles have the angle:

[tex]\angle DBE\text{ which is the same as }\angle ABC[/tex]

We can say that the triangles ABC and DBE are similar by the Theorem SAS (side-angle-side):

[tex]\begin{gathered} \text{Side 1: }AB\approx DB \\ \text{Angle: }\angle DBE \\ \text{Side 2: }CB\approx EB \end{gathered}[/tex]

Thus, their sides are proportional, and their corresponding angles are congruent.

As the angles ACE and DEB are corresponding (on the two triangles), they have the same measure:

[tex]\begin{gathered} m\angle ACE=m\angle DEB \\ 101-5x=119-7x \\ \text{And we solve for x:} \\ 101-5x+7x=119-7x+7x \\ 101+2x=119 \\ 2x=119-101 \\ 2x=18 \\ x=\frac{18}{2}=9 \end{gathered}[/tex]

This means that the value of x is 9. Now, replacing on the value of ACE, and we get:

[tex]\begin{gathered} m\angle ACE=101-5x \\ =101-5(9) \\ =101-45 \\ =56 \end{gathered}[/tex]

Thus, the value of the angle ∠ACE is 56°.