Question

math 2
unit 6 - triangles & congruence
lesson 1 - congruent triangles & cpctc

review: similar triangles are the same shape but different sizes. in order for two triangles to be similar, the corresponding angles must be congruent and the corresponding sides must be proportional.

congruent triangles: triangles that are the same

• each triangle has three congruent
• if all six of the corresponding parts of two triangles are

definition of congruent triangles (cpctc):

• two triangles are congruent if and only if their corresponding parts are
• cpctc - corresponding parts of congruent triangles are congruent

1. write a congruency statement for the two triangles at right.
2. list all of the congruent parts if △efg ≅ △hgf.
3. △wxy ≅ △zyx solve for p and q.

Explanation:

Step1: Recall congruent - triangle definitions

Congruent triangles have same shape and size. Each has three congruent angles and three congruent sides. If all six corresponding parts are congruent, the triangles are congruent. Definition of CPCTC: Corresponding parts of congruent triangles are congruent.

Step2: Write congruency statement for first pair

Since corresponding sides and angles are marked congruent, the congruency statement for the triangles with vertices \(A\), \(C\), \(R\) and \(E\), \(G\), \(O\) is \(\triangle ACR\cong\triangle EGO\).

Step3: List congruent parts for \(\triangle EFG\cong\triangle HGF\)

Angles: \(\angle E\cong\angle H\), \(\angle F\cong\angle G\), \(\angle G\cong\angle F\); Sides: \(EF\cong HG\), \(FG\cong GF\), \(EG\cong HF\).

Step4: Solve for \(p\) and \(q\) in \(\triangle WXY\cong\triangle ZYX\)

Since corresponding sides are equal, for side - length equality:

For \(p\):

We have \(7p + 13=20\).
Subtract 13 from both sides: \(7p=20 - 13\), so \(7p = 7\).
Divide both sides by 7: \(p = 1\).

For \(q\):

We have \(2q+3 = 11\).
Subtract 3 from both sides: \(2q=11 - 3\), so \(2q = 8\).
Divide both sides by 2: \(q = 4\).

Answer:

1. \(\triangle ACR\cong\triangle EGO\)
2. Angles: \(\angle E\cong\angle H\), \(\angle F\cong\angle G\), \(\angle G\cong\angle F\); Sides: \(EF\cong HG\), \(FG\cong GF\), \(EG\cong HF\)
3. \(p = 1\), \(q = 4\)