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In the last lesson, we learned a rule for determining that two triangles are congruent without having to find a rigid motion which connects them: the Side-Angle-Side or SAS rule. In this lesson, we will look at some other rules for determining whether two figures are congruent.
Two triangles $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$ are drawn to the left, along with controls which move the red triangle $▵ABC$. Notice that the angles $\cl"red"{∠A}$ and $\cl"blue"{∠A'}$ are equally large, as are the angles $\cl"red"{∠B}$ and $\cl"blue"{∠B'}$ and the sides $\cl"red"\ov{AB}$ and $\cl"blue"\ov{A'B'}$ which join them. We’ve also drawn rays from $\cl"red"A$ and $\cl"red"B$ through $\cl"red"C$, and from $\cl"blue"{A'}$ and $\cl"blue"{B'}$ through $\cl"blue"{C'}$.
If you have two triangles with corresponding measurements equal for two of their angles and for the side between those two angles, as in $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$, then you can always make the triangles coincide with a rigid motion in this way:
This gives another rule which lets you see if two triangles are congruent. It is called the Angle-Side-Angle or ASA rule for congruence of triangles.
A triangle $\cl"red"{▵ABC}$ is drawn to the left, along with the rays from $\cl"red"A$ through $\cl"red"C$ and from $\cl"red"B$ through $\cl"red"C$. By sliding the sliders, you can change the lengths of the side $\cl"red"\ov{AB}$ and the measure of its surrounding angles $\cl"red"{∠A}$ and $\cl"red"{∠B}$. The rays show all the possible locations of $\cl"red"C$ for a given measurement of $\cl"red"{∠A}$ or $\cl"red"{∠B}$. The approximate length of each side and measure of each angle of the triangle are also shown.
The SAS and ASA rules for triangle congruence involve measuring both angles and sides. Now we’ll look at a rule where only sides are measured.
Two triangles $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$ are drawn to the left, along with controls which move the red triangle $▵ABC$. Notice that the sides $\cl"red"\ov{AB}$ and $\cl"blue"\ov{A'B'}$ are equally long, as are the sides $\cl"red"\ov{AC}$ and $\cl"blue"\ov{A'C'}$, and the sides $\cl"red"\ov{BC}$ and $\cl"blue"\ov{B'C'}$. We’ve also drawn circles centered at the points $\cl"red"A$ and $\cl"red"B$ passing through the point $\cl"red"C$, and circles centered at the points $\cl"blue"{A'}$ and $\cl"blue"{B'}$ passing through the point $\cl"blue"{C'}$.
If you have two triangles with all three corresponding side measurements equal, as in $\cl"red"{▵ABC}$ and $\cl"blue"{▵A'B'C'}$, then you can always make the triangles coincide with a rigid motion in this way:
This gives you the Side-Side-Side or SSS rule for congruence of triangles.
A triangle $\cl"red"{▵ABC}$ is drawn to the left, along with the circles centered at $\cl"red"A$ and $\cl"red"B$ passing through $\cl"red"C$. By sliding the sliders, you can change the lengths of the sides of the triangle. The circles show all the possible locations of $\cl"red"C$ for a given measurement of $\cl"red"\ov{AC}$ or $\cl"red"\ov{BC}$. The approximate length of each side and measure of each angle of the triangle are also shown.
The SSS rule lets you see that two triangles are congruent by only looking at their sides. Let’s see if there could be a way to see that two triangles are congruent by only looking at their angles.
The two triangles drawn to the left have the same angle measurements as each other, for all their angles. (That is, $\cl"red"{∠A}$ and $\cl"blue"{∠A'}$ have the same measurements, as do $\cl"red"{∠B}$ and $\cl"blue"{∠B'}$, and $\cl"red"{∠C}$ and $\cl"blue"{∠C'}$.)
Because we can find triangles which have the same angle measurements but are not congruent, there can be no Angle-Angle-Angle (AAA) congruence rule for triangles.
A quadrilateral (four-sided figure) $\cl"red"{ABCD}$ is drawn to the left. The slider below the quadrilateral changes the measure of $\cl"red"{∠A}$.
For figures that are not triangles, it is difficult to come up with a congruence rule like the ones from this lesson. Usually, if you want to show that two non-triangular figures are congruent, you need to find a rigid motion which makes them coincide.