Which Shows Two Triangles That Are Congruent by ASA?
In geometry, two triangles are congruent if they have the same shape and size. There are five different ways to prove that two triangles are congruent: SSS, SAS, ASA, AAS, and HL.
SSS stands for side-side-side. Two triangles are congruent by SSS if the corresponding sides of each triangle are congruent.
SAS stands for side-angle-side. Two triangles are congruent by SAS if two corresponding sides of each triangle are congruent and the included angle between those sides is congruent.
ASA stands for angle-side-angle. Two triangles are congruent by ASA if two corresponding angles of each triangle are congruent and the included side between those angles is congruent.
AAS stands for angle-angle-side. Two triangles are congruent by AAS if two corresponding angles of each triangle are congruent and a non-included side of each triangle is congruent.
HL stands for hypotenuse-leg. Two right triangles are congruent by HL if the hypotenuse and a leg of each triangle are congruent.
Which Shows Two Triangles That Are Congruent by ASA?
The answer to this question is the following:
Two triangles are congruent by ASA if two corresponding angles of each triangle are congruent and the included side between those angles is congruent.
For example, the following two triangles are congruent by ASA:
[asy] pair A,B,C,D,E,F; A = (0,0); B = (60,0); C = (30,45); D = (100,0); E = (160,0); F = (130,45); draw(A--B--C--cycle); draw(D--E--F--cycle); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,S); label("$E$",E,S); label("$F$",F,N); label("$50^\circ$",(B+C)/2,S); label("$50^\circ$",(E+F)/2,S); label("$60$",(A+C)/2,N); label("$60$",(D+F)/2,N); [/asy] In this example, $\angle ABC = \angle DEF = 50^\circ$ and $AC = DF = 60$. Therefore, the two triangles are congruent by ASA.
Additional Questions
Here are some additional questions that you may be asked about ASA congruence:
- What are the two corresponding angles that must be congruent in order for two triangles to be congruent by ASA?
The answer is that the two corresponding angles must be included by the included side.
- What is the included side?
The included side is the side that is between the two corresponding angles.
- Can two triangles be congruent by ASA if only one of the two corresponding angles is congruent?
The answer is no. In order for two triangles to be congruent by ASA, both of the two corresponding angles must be congruent.
- Can two triangles be congruent by ASA if the included side is not congruent?
The answer is no. In order for two triangles to be congruent by ASA, the included side must be congruent.
