Draw two line segments and of two different lengths, say and respectively. Then, at the points and respectively, construct angles and of some measures, say, and . Also, at the points and , construct angles and of and respectively (see Fig. 6.23).

Understand the Question The sum of all interior angles in any triangle is always . When constructing triangles given two angles, the third angle is uniquely determined by the angle sum property. AAA Similarity Criterion: If corresponding angles of two triangles are equal, the triangles are similar. For two triangles to be congruent, their corresponding side lengths must also be equal. Step 1 · Find in In , and . By the angle sum property of a triangle Step 2 · Find in In , and . By the angle sum property of a triangle Step 3 · Compare the Triangles Comparing and : Corresponding angles are equal: , , and . By the AAA similarity criterion, . Since corresponding sides have different lengths (), the triangles are not congruent. Answer > and . The triangles and are similar, but not congruent. Common Mistakes Confusing Similarity with Congruence: Having all three corresponding angles equal (AAA) guarantees similarity, not congruence. Equal side lengths are necessary for congruence. Angle Sum Errors: Making arithmetic mistakes when calculating the third angle from .

Triangles | Activity

Question 4

Draw two line segments $BC$ and $EF$ of two different lengths, say $3\text{ cm}$ and $5\text{ cm}$ respectively. Then, at the points $B$ and $C$ respectively, construct angles $\angle PBC$ and $\angle QCB$ of some measures, say, $60^\circ$ and $40^\circ$ . Also, at the points $E$ and $F$ , construct angles $\angle REF$ and $\angle SFE$ of $60^\circ$ and $40^\circ$ respectively (see Fig. 6.23).

Solution

Understand the Question

The sum of all interior angles in any triangle is always $180^\circ$ .
When constructing triangles given two angles, the third angle is uniquely determined by the angle sum property.
AAA Similarity Criterion: If corresponding angles of two triangles are equal, the triangles are similar.
For two triangles to be congruent, their corresponding side lengths must also be equal.

Step 1 · Find $\angle A$ in $\triangle ABC$

In $\triangle ABC$ , $\angle B = 60^\circ$ and $\angle C = 40^\circ$ . Diagram 1

By the angle sum property of a triangle

\begin{aligned} \angle A + \angle B + \angle C &= 180^\circ \\ \angle A + 60^\circ + 40^\circ &= 180^\circ \\ \angle A + 100^\circ &= 180^\circ \\ \angle A &= 180^\circ - 100^\circ = 80^\circ \end{aligned}

Step 2 · Find $\angle D$ in $\triangle DEF$

In $\triangle DEF$ , $\angle E = 60^\circ$ and $\angle F = 40^\circ$ . Diagram 2

By the angle sum property of a triangle

\begin{aligned} \angle D + \angle E + \angle F &= 180^\circ \\ \angle D + 60^\circ + 40^\circ &= 180^\circ \\ \angle D + 100^\circ &= 180^\circ \\ \angle D &= 180^\circ - 100^\circ = 80^\circ \end{aligned}

Step 3 · Compare the Triangles

Comparing $\triangle ABC$ and $\triangle DEF$ :

Corresponding angles are equal: $\angle A = \angle D = 80^\circ$ , $\angle B = \angle E = 60^\circ$ , and $\angle C = \angle F = 40^\circ$ .
By the AAA similarity criterion, $\triangle ABC \sim \triangle DEF$ .
Since corresponding sides have different lengths ( $BC = 3\text{ cm} \neq EF = 5\text{ cm}$ ), the triangles are not congruent.

Answer

$\angle A = 80^\circ$ and $\angle D = 80^\circ$ . The triangles $\triangle ABC$ and $\triangle DEF$ are similar, but not congruent.

Common Mistakes

Confusing Similarity with Congruence: Having all three corresponding angles equal (AAA) guarantees similarity, not congruence. Equal side lengths are necessary for congruence.
Angle Sum Errors: Making arithmetic mistakes when calculating the third angle from $180^\circ - (\angle B + \angle C)$ .