Question

For each pair of congruent triangles, name the congruent angles and sides. Then draw the triangles, using arcs and slash marks to show the congruent angles and sides. $$\triangle Q R S \cong \triangle T U V$$

Short answer

Congruent angles: \(\angle Q \cong \angle T\), \(\angle R \cong \angle U\), \(\angle S \cong \angle V\). Congruent sides: \(QR \cong TU\), \(RS \cong UV\), \(SQ \cong VT\).

Step-by-step solution

Identify Congruent Angles

In congruent triangles, corresponding angles are congruent. For \(\triangle QRS \cong \triangle TUV\), the angles are congruent as follows: \(\angle Q \cong \angle T\), \(\angle R \cong \angle U\), and \(\angle S \cong \angle V\).

Identify Congruent Sides

In congruent triangles, corresponding sides are congruent. Therefore, the sides are congruent as follows: \(QR \cong TU\), \(RS \cong UV\), and \(SQ \cong VT\).

Draw the Triangles

Draw two triangles. Label one as \(\triangle QRS\) and the other as \(\triangle TUV\). For each pair of congruent angles, use the same number of arcs to show congruence (e.g., one arc for \(\angle Q\) and \(\angle T\), two arcs for \(\angle R\) and \(\angle U\), and so on).

Mark the Congruent Sides with Slash Marks

On the triangles, mark corresponding congruent sides with the same number of slash marks. For example, one slash mark for \(QR\) and \(TU\), two slash marks for \(RS\) and \(UV\), and three slash marks for \(SQ\) and \(VT\).

Key concepts

Corresponding Angles

When discussing congruent triangles, corresponding angles are key components to understand. Corresponding angles are pairs of angles that hold the same relative position in two different shapes. Imagine two triangles that are mirror images. These triangles will each have three angles, and these angles are said to be corresponding if they are in the same position on each triangle. For instance, in triangles \( \triangle QRS \) and \( \triangle TUV \), the corresponding angles are: \( \angle Q \) is to \( \angle T \), \( \angle R \) is to \( \angle U \), and \( \angle S \) is to \( \angle V \).

• Their correspondency means that corresponding angles are equal in measure, which reflects the concept of congruence.
• This equality is crucial in proving two triangles are congruent.

Being diligent in identifying these angles correctly helps in accurately recognizing congruent triangles.

Corresponding Sides

In the world of congruent triangles, corresponding sides work just like corresponding angles. They are respective pairs of sides from each triangle that sit in matching positions. Think of them like twinned line segments. When you have two triangles, the sides are said to be corresponding if they fit into the same respective places in both triangles. For example, notice in triangles \( \triangle QRS \) and \( \triangle TUV \):

• Side \( QR \) corresponds to side \( TU \).
• Side \( RS \) corresponds to side \( UV \).
• Side \( SQ \) corresponds to side \( VT \).

• Corresponding sides mean they are equal in length.
• This is one of the conditions that define two triangles as congruent.

Identifying corresponding sides accurately ensures that the triangles are truly congruent and not just similar.

Triangle Congruence

Triangle congruence is the overarching concept that binds everything together when studying pairs of triangles. For two triangles to be congruent, they must meet specific criteria, including having corresponding angles and sides that are identical in measure or length.

• To prove congruence, you can use congruence statements like \( \triangle QRS \cong \triangle TUV \), which tells us these triangles are congruent.
• There's a range of criteria to check for congruence: Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Side-Side-Side (SSS), each serving different configurations.

This assures that every element of one triangle matches perfectly with its counterpart in the other, both in terms of angles' measurement and sides' length. Triangle congruence encapsulates these elements into a simple yet profound geometric principle.