Lesson 11

Which of the following criteria always proves triangles congruent? Select all that apply. 

3 congruent angles

3 congruent sides

Corresponding congruent Side-Angle-Side 

Corresponding congruent Side-Side-Angle

Corresponding congruent Angle-Side-Angle

Here are some measurements for triangle \(\ ABC \) and triangle \(XYZ\):

• Angle \( ABC\) and angle \(XYZ \) are both 30°
• \(BC\) and \(YZ\) both measure 6 units
• \(CA\) and \(ZX\) both measure 4 units

Lin thinks thinks these triangles must be congruent. Priya says she knows they might not be congruent. Construct 2 triangles with the given measurements that aren't congruent. Explain why triangles with 3 congruent parts aren't necessarily congruent.

Angle \(CBE\) is congruent to angle \(DAE\).

Angle \(CEB\) is congruent to angle \(DEA\).

Segment \(DA\) is congruent to segment \(CB\).

Segment \(DC\) is congruent to segment \(AB\).

Line \(DC\) is parallel to line \(AB\).

Line \(DA\) is parallel to line \(CB\).

Andre is thinking through a proof using a reflection to show that a triangle is isosceles given that its base angles are congruent. Complete the missing information for his proof. 

Construct \(AB\) such that \(AB\) is the perpendicular bisector of segment \(CD\). We know angle \(ADB\) is congruent to \(\underline{\hspace{.5in}1\hspace{.5in}}\). \(DB\) is congruent to \(\underline{\hspace{.5in}2\hspace{.5in}}\) since \(AB\) is the perpendicular bisector of \(CD\).  Angle \(\underline{\hspace{.5in}3\hspace{.5in}}\) is congruent to angle \(\underline{\hspace{.5in}4\hspace{.5in}}\) because they are both right angles. Triangle \(ABC\) is congruent to triangle \(\underline{\hspace{.5in}5\hspace{.5in}}\) because of the \(\underline{\hspace{.5in}6\hspace{.5in}}\) Triangle Congruence Theorem. \(AD\) is congruent to \(\underline{\hspace{.5in}7\hspace{.5in}}\) because they are corresponding parts of congruent triangles. Therefore, triangle \(ADC\) is an isosceles triangle.

Translate \(DEF\) using directed line segment \(EA\). Rotate \(D’E’F’\) using \(A\) as the center so that \(D’\) coincides with \(C\). Reflect \(D’’E’’F’’\) across line \(AC\).

Translate \(DEF\) using directed line segment \(EA\). Rotate \(D’E’F’\) using \(A\) as the center so that \(D’\) coincides with \(C\). Reflect \(D’’E’’F’’\) across line \(AB\).

Translate \(DEF\) using directed line segment \(EA\). Rotate \(D’E’F’\) using \(A\) as the center so that \(D’\) coincides with \(B\). Reflect \(D’’E’’F’’\) across line \(AC\).

Translate \(DEF\) using directed line segment \(EA\). Rotate \(D’E’F’\) using \(A\) as the center so that \(D’\) coincides with \(B\). Reflect \(D’’E’’F’’\) across line \(AB\).