Determine whether the triangles DCE and ABE congruent?
If so, according to which congruence theorem?
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Determine whether the triangles DCE and ABE congruent?
If so, according to which congruence theorem?
Congruent triangles are triangles that are identical in size, meaning that if we place one on top of the other, they will match exactly.
In order to prove that a pair of triangles are congruent, we need to prove that they satisfy one of these three conditions:
SSS - Three sides of both triangles are equal in length.
SAS - Two sides are equal between the two triangles, and the angle between them is equal.
ASA - Two angles in both triangles are equal, and the side between them is equal.
If we take an initial look at the drawing, we can already observe that there is one equal side between the two triangles, as they are both marked in blue,
We don't have information regarding the other sides, thus we can rule out the first two conditions,
And now we'll focus on the last condition - angle, side, angle.
We can observe that angle D equals angle A, both equal to 50 degrees,
Let's proceed to the angles E.
At first glance, one might think that there's no way to know if these angles are equal, however if we look at how the triangles are positioned,
We can see that these angles are actually corresponding angles, and corresponding angles are of course equal.
Therefore - if the angle, side, and second angle are equal, we can prove that the triangles are equal using the ASA condition
Congruent according to A.S.A
Look at the triangles in the diagram.
Which of the statements is true?
Look carefully at the diagram! Both triangles share the side AE (or DE depending on how you read it). When triangles overlap like this, they often share a common side or vertex.
Vertical angles are opposite angles formed when two lines intersect. They're always equal because they're supplementary to the same angles. In this problem, angles at point E are vertical angles.
We don't have enough information about the side lengths. We only know about the angles (50° and the vertical angles) and one shared side. ASA is perfect when you have angle-side-angle information!
Match angles by their position in each triangle: angle D corresponds to angle A (both 50°), the angles at E are vertical angles, and the shared side connects the triangles.
Without the 50° markings, you couldn't prove congruence! Those angle measurements are crucial evidence. Always look for given angle measures, parallel line markings, or other geometric clues in the diagram.
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