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Problem 120

LET \(A B C D\) be a quadrilateral in which \(A D=B C\) and \(\mathrm{AB}=\mathrm{CD}\). Let its diagonals, \(\underline{\mathrm{AC}}\) and \(\underline{\mathrm{BD}}\), intersect at point E. a) Prove that $\triangle \mathrm{ABC} \cong \Delta \mathrm{CDA} ; \mathrm{b}\( ) Prove that \)\angle \mathrm{DAC} \cong \angle \mathrm{BCA}$; c) Prove that \(\triangle \mathrm{ABD} \cong \Delta \mathrm{CDB} ; \mathrm{d}\) ) Prove that \(\angle \mathrm{ADB} \cong \angle \mathrm{CBD}\).

Expert verified

In summary, we proved that:
a) \(\triangle ABC \cong \triangle CDA\) using the SSS congruence postulate.
b) \(\angle DAC \cong \angle BCA\) as corresponding angles of congruent triangles.
c) \(\triangle ABD \cong \triangle CDB\) using the SAS congruence postulate.
d) \(\angle ADB \cong \angle CBD\) as corresponding angles of congruent triangles.

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Chapter 8

$\underline{\mathrm{DB}} \cong \underline{\mathrm{EA}}, \underline{\mathrm{AD}} \cong \underline{\mathrm{BE}} .\( Prove: \)\angle \mathrm{DAB} \cong \angle \mathrm{EBA}$

Chapter 8

Let \(\mathrm{ABC}\) be an equilateral triangle. Let \(\mathrm{D}\) be the midpoint of \(\underline{A B}\). Prove that $\triangle D C B \cong \triangle D C A\(. What kind of triangle is \)\triangle \mathrm{DCB}$ ? What can be said about \(\angle \mathrm{ACD}\) and \(\angle \mathrm{BCD} ?\) (See Figure.)

Chapter 8

Given: QS intersects \(\underline{P R}\) at \(\mathrm{T}\) such that \(\mathrm{RQ}=\mathrm{RS}\) and \(\mathrm{QT}=\mathrm{ST}\) Prove: \(\underline{\text { TP }}\) bisects \(\angle \mathrm{SPQ}\).

Chapter 8

Let \(\triangle \mathrm{ABC}\) be an equilateral triangle and let \(\mathrm{D}\) be the midpoint of \(\underline{A B}\). In \(\triangle D C B\), what are the measures of \(\angle B D C\), \(\angle \mathrm{DCB}\), and \(\angle \mathrm{DBC}\) ? If \(\mathrm{BC}=2\), what does \(\mathrm{DB}\) equal?

Chapter 8

Given: $\underline{C A} \cong \underline{D B} \cdot \underline{C B} \cong \underline{D A} .\( Prove \)\triangle A B C \cong \triangle B A D$.

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