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Why are triangles
and
congruent?
A triangle ABC is bisected by a vertical line AD to form two right-angle triangles. In which CD and BD are congruent, and AC and AB are congruent
Triangle
is congruent to triangle
because
triangle
maps it onto triangle
.
Triangle PQR is congruent to triangle XYZ because triangle XYZ has the same length sides and angles as triangle PQR.
Triangle ABC is congruent to triangle DEF because triangle DEF has the same length sides and angles as triangle ABC.
Triangle ACD is congruent to triangle ABD because side CD is congruent to side BD and side AC is congruent to side AB.
To determine why triangles and are congruent, we need to look for corresponding congruent parts or apply one of the congruence criteria.
In this example, we are given that triangle ABC is bisected by a vertical line AD to form two right-angle triangles. Let's label the vertices of triangle ABC as A, B, and C. Triangle ABC is divided by the vertical line AD, creating two smaller right-angle triangles: triangle ACD and triangle ABD.
We are also given that CD and BD are congruent (or equal in length), and AC and AB are congruent.
To show that triangles ACD and ABD are congruent, we can use the Side-Angle-Side (SAS) congruence criterion. According to this criterion, if two triangles have congruent corresponding sides and the included angle between those sides is congruent, then the triangles are congruent.
In our case, we have CD ≅ BD (the sides are congruent or equal) and AC ≅ AB (the sides are congruent or equal). The angle ∠CAD is congruent to ∠BAD because they are corresponding angles formed by the vertical line AD and horizontal line BC, which are intersecting lines.
So, the answer is:
Triangle ACD is congruent to triangle ABD because they have congruent corresponding sides (CD ≅ BD) and the included angle (∠CAD ≅ ∠BAD).
It is important to understand and apply the properties and congruence criteria for triangles to determine why they are congruent.