Before learning the ASA formula, let us recall what is congruence. If two triangles are congruent it means that three sides of one triangle will be (respectively) equal to the three sides of the other and three angles of one triangle will be (respectively) equal to the three angles of the other. We need not know all the angles and sides of two triangles to determine if they are congruent. We can simply use the ASA formula.
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ASA formula is one of the criteria used to determine congruence. ASA congruence criterion states that
"if two angles of one triangle, and the side contained between these two angles, are respectively equal to two angles of another triangle and the side contained between them, then the two triangles will be congruent".
Let us see the applications of the ASA formula in the following section.
Solved Examples Using ASA Formula
Example 1: In an isosceles triangle ABC, AB = BC. A perpendicular bisector is drawn from A on BC. What is the relation between triangle ABD and ACD?
To find: Relation between triangle ABD and ACD.
AD is the perpendicular bisector (given)
Since AD is the perpendicular bisector, therefore angle ADB = angle ADC = 90º
Since AD is the bisector, therefore BD = DC
For isosceles triangle ABC,
angle ABC = angle ACB
Therefore by ASA formula, triangle ABD and ACD are congruent.
Answer: Triangle ABD and ACD are congruent by ASA congruency.
Example 2: If two triangles ABC and PQR are congruent by ASA formula. If two angles of triangle ABC are 60º and 40º. Find the measures of all the angles of triangle PQR.
To find: Angles of PQR.
ABC is congruent to PQR(given)
Two angles of the triangle ABC are 60º and 40º,
Then the third angle will be = 180-(60 +40) = 80º
As triangle ABC and PQR are congruent, then the measure of all the angles will be congruent to each other.
Then the measure of all the angles of triangle PQR is 60º, 40º, and 80º.
Answer: Then the measure of all the angles of triangle PQR is 60º, 40º, and 80º.