Unit 2, Lesson 7
Angle-Side-Angle Triangle Congruence
Geometry
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Assertion: Through 2 distinct points passes a unique line. Two lines are said to be distinct if there is at least 1 point that belongs to one but not the other. Otherwise, we say the lines are the same. Lines that have no point in common are said to be parallel.
Conclusion: Given two distinct lines, either they are parallel, or they have exactly 1 point in common.
What do you notice? What do you wonder?
Lines and are parallel. . Find , , , , , , and .
Parallel lines l and m, intersected by a diagonal line running down and to the left. Angles A through H are marked. On the left side of the diagonal, starting above line l and moving downward, angles are marked as follows: B, C, F, G. On the right side of the diagonal, starting above line L and moving downward, angles are marked as follows: A equals 42 degrees, D, E, H. Circles are drawn around angles A, B C and D and angles E, F, G and H.
We know that in two triangles, if 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent, then the triangles must be congruent. But we don’t always know that 2 pairs of corresponding sides are congruent. For example, when proving that opposite sides are congruent in any parallelogram, we only have information about 1 pair of corresponding sides. That is why we need ways other than the Side-Angle-Side Triangle Congruence Theorem to prove triangles are congruent.
In two triangles, if 2 pairs of corresponding angles and the pair of corresponding sides between the angles are congruent, then the triangles must be congruent. This is called the Angle-Side-Angle Triangle Congruence Theorem.
When proving that two triangles are congruent, look at the diagram and given information, and think about whether it will be easier to find 2 pairs of corresponding angles that are congruent or 2 pairs of corresponding sides that are congruent. Then check if there is enough information to use the Angle-Side-Angle Triangle Congruence Theorem or the Side-Angle-Side Triangle Congruence Theorem.
The Angle-Side-Angle Triangle Congruence Theorem can be used to prove that, in a parallelogram, opposite sides are congruent. A parallelogram is defined to be a quadrilateral in which the 2 pairs of opposite sides are parallel.
We could prove that triangles and are congruent by the Angle-Side-Angle Triangle Congruence Theorem. Then we can say segment is congruent to segment because they are corresponding parts of congruent triangles.