Question

1. given \\( \triangle a b c \cong \triangle d e f \\), name three pairs of congruent sides.
2. given \\( \triangle m n o \cong \triangle p q r \\), name three pairs of congruent angles.

in the diagram, \\( \triangle m k l \cong \triangle j e t \\). complete the statement.
given: \\( m \angle k = 32 ^ { \circ } \\& \angle j = 85 ^ { \circ } \\)

1. \\( \angle l \cong \\)
2. \\( \overline { m k } \cong \\)
3. \\( m \angle m = \\)
4. \\( m \angle t = \\)
5. \\( m l = \\)
6. \\( \triangle e t j \cong \\)

complete this statement.

1. if \\( \triangle w r d \cong \triangle p l k , \overline { w r } \\) then \\( \cong \\)
2. if \\( \triangle b g t \cong \triangle d s n \\), then \\( \angle t \cong \\)
3. if \\( \triangle s v p \cong \triangle m t q \\), then \\( \overline { p s } \cong \\)
4. if \\( \triangle j c x \cong \triangle m w p \\), then \\( \overline { x c } \cong \\)
5. if \\( \triangle r h k \cong \triangle w v o \\), then \\( \triangle k r h \cong \\)
6. if \\( \triangle p m c \cong \triangle l d x \\), then \\( \angle m \cong \\)

determine if the triangles are congruent by sss, sas, asa, aas, hl or not enough information. make sure to label all the information given or deduced by making corresponding marks on the sides/angles of the triangles. dont prove.
15)

1. if \\( t r \cong q r \\).

\\( \triangle \cong \triangle \\)
\\( \triangle \cong \triangle \\)
17)

1. if \\( g f \cong b c \\).

\\( \triangle \cong \triangle \\)
\\( \triangle \cong \triangle \\)

Explanation:

Step1: Use congruent triangle properties

For congruent triangles, corresponding parts are congruent. For \(\triangle ABC\cong\triangle DEF\), the order of the letters gives the correspondence. So \(AB = DE\), \(BC=EF\), \(AC = DF\).

Step2: For \(\triangle MNO\cong\triangle PQR\)

Using the correspondence of the letters, \(\angle M=\angle P\), \(\angle N=\angle Q\), \(\angle O=\angle R\).

Step3: For \(\triangle MKL\cong\triangle JET\)

• \(\angle L\cong\angle T\) (corresponding angles)
• In \(\triangle JET\), \(\angle E=180^{\circ}- 85^{\circ}-32^{\circ}=63^{\circ}\), so \(m\angle M = 85^{\circ}\) (corresponding to \(\angle J\))
• \(ML = JE\) (corresponding sides)
• \(\overline{MK}\cong\overline{JE}\) (corresponding sides)
• \(m\angle T=32^{\circ}\) (corresponding to \(\angle K\))
• \(\triangle ETJ\cong\triangle MLK\) (by correspondence)

Step4: For other congruent triangle statements

• If \(\triangle WRD\cong\triangle PLK\), \(\overline{WR}\cong\overline{PL}\)
• If \(\triangle BGT\cong\triangle DSN\), \(\angle T\cong\angle N\)
• If \(\triangle SVP\cong\triangle MTQ\), \(\overline{PS}\cong\overline{QT}\)
• If \(\triangle JCX\cong\triangle MWP\), \(\overline{XC}\cong\overline{WP}\)
• If \(\triangle RHK\cong\triangle WVO\), \(\triangle KRH\cong\triangle OWV\)
• If \(\triangle PMC\cong\triangle LDX\), \(\angle M\cong\angle D\)

Step5: Congruence criteria

• For 15), \(\triangle EDC\cong\triangle EBC\) (by AAS, as two angles and a non - included side are congruent)
• For 16), \(\triangle TSR\cong\triangle UQR\) (by AAS, given \(\angle T=\angle U\), \(\angle SRT=\angle QRU\), \(TR = QR\))
• For 17), \(\triangle HTA\cong\triangle PMA\) (by AAS, vertical angles are equal, and two other angles)
• For 18), \(\triangle EFG\cong\triangle BCD\) (by SAS, \(EF = BC\), \(\angle F=\angle C = 90^{\circ}\), \(FG=CD\))

Answer:

1. \(AB = DE\), \(BC = EF\), \(AC=DF\)
2. \(\angle M=\angle P\), \(\angle N=\angle Q\), \(\angle O=\angle R\)
3. \(\angle T\)
4. \(85\)
5. \(JE\)
6. \(\overline{JE}\)
7. \(32\)
8. \(\triangle MLK\)
9. \(\overline{PL}\)
10. \(\angle N\)
11. \(\overline{QT}\)
12. \(\overline{WP}\)
13. \(\triangle OWV\)
14. \(\angle D\)
15. \(\triangle EDC\cong\triangle EBC\)
16. \(\triangle TSR\cong\triangle UQR\)
17. \(\triangle HTA\cong\triangle PMA\)
18. \(\triangle EFG\cong\triangle BCD\)