Feature
Use the following information for questions 2–5. Questions 2–4 are the steps in a proof, and question 5 is the conclusion to that proof.
Given: triangleABD congruent to triangleCDB, modifying above upper A upper B with bar||modifying above Upper C Upper D with barA parallelogram is shown. Its points from bottom left moving clockwise are A, B, C, and D. Two diagonal lines go from point A to C and point B to D. The intersection of the lines in the middle is marked point E.
Prove:triangleABE congruent to triangleCDE
Question
A student writes the statement angleBEA congruent to angleDEC to help prove the triangles are congruent. What reason should the student give?
(1 point)
Responses
Given
Given
Angles cut by a bisector are congruent.
Angles cut by a bisector are congruent.
Base angles of an isosceles triangle are congruent.
Base angles of an isosceles triangle are congruent.
Vertical angles are congruent.
Vertical angles are congruent.
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Vertical angles are congruent.
The reason the student should give is: "Vertical angles are congruent."
To determine the reason the student should give for the statement "angle BEA congruent to angle DEC" in order to prove the triangles congruent, we need to consider the given information and the properties of the diagram.
In the given diagram, we have a parallelogram with points A, B, C, and D labeled in a clockwise direction. Two diagonal lines, one from point A to C and one from point B to D, intersect at point E in the middle.
To prove that triangle ABE is congruent to triangle CDE, we need to provide a reason that supports the statement "angle BEA congruent to angle DEC."
Looking at the diagram, we can observe that angle BEA and angle DEC are formed by the intersection of the diagonal lines from A to C and B to D. These angles are formed by a bisector (the diagonal line) dividing the larger angles (angle BAD and angle BCD) into equal parts.
Therefore, the reason the student should give is "Angles cut by a bisector are congruent." This property states that when a bisector cuts through two angles, the resulting angles on opposite sides of the bisector are congruent.
So, to prove the congruence of triangles ABE and CDE, one of the steps in the proof is to assert that angle BEA is congruent to angle DEC, based on the property that angles cut by a bisector are congruent.