Lesson 11

Side-Side-Angle (Sometimes) Congruence

  • Let’s explore triangle congruence criteria that are ambiguous.

Problem 1

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

A:

3 congruent angles

B:

3 congruent sides

C:

Corresponding congruent Side-Angle-Side 

D:

Corresponding congruent Side-Side-Angle

E:

Corresponding congruent Angle-Side-Angle

Problem 2

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.

Problem 3

Jada states that diagonal \(WY\) bisects angles \(ZWX\) and \(ZYX\). Is she correct? Explain your reasoning,

Triangle WYZ and WYX. WZ is congruent to WX and YZ is congruent to YX. The triangles share side WY.
(From Unit 2, Lesson 9.)

Problem 4

Select all true statements based on the diagram.

Quadrilateral ABCD. Line AB is parallel to line DC, both cut by congruent transversals AD and BC. Diagonals AC and DB bisect each other at point E at a right angle. Segment AE is congruent to segment BE.
A:

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

B:

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

C:

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

D:

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

E:

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

F:

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

(From Unit 2, Lesson 10.)

Problem 5

\(WXYZ\) is a kite. Angle \(WXY\) has a measure of 94 degrees and angle \(ZWX\) has a measure of 112 degrees. Find the measure of angle \(ZYW\).

Kite W X Y Z. Diagonal W Y is drawn in. Sides Z W and W X have single tick marks. Sides X Y and Y Z have double tick marks.
(From Unit 2, Lesson 9.)

Problem 6

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. 

Triangle A C D. Angles D and C have arcs with single tick marks. Point B lies on side D C, and segment A B is drawn in.

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.

(From Unit 2, Lesson 8.)

Problem 7

The triangles are congruent. Which sequence of rigid motions takes triangle \(DEF\) onto triangle \(BAC\)?

Triangle ABC is congruent to triangle EDF.
A:

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 \(AC\).

B:

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 \(AB\).

C:

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 \(AC\).

D:

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 \(AB\).

(From Unit 2, Lesson 3.)