1. Proof:
- Given: $\overline{BD} \perp \overline{AC}$, so $\angle BDA$ and $\angle BDC$ are right angles.
- Given: D is the midpoint of $\overline{AC}$, so $\overline{AD} \cong \overline{CD}$.
- $\overline{BD} \cong \overline{BD}$ by the Reflexive Property.
- Therefore, $\triangle ABD \cong \triangle CBD$ by SAS (Side-Angle-Side).
- By CPCTC, $\angle A \cong \angle C$.
2. Proof:
- Given: $\angle E \cong \angle P$.
- Given: K is the midpoint of $\overline{EP}$, so $\overline{EK} \cong \overline{PK}$.
- $\angle EKG \cong \angle PKM$ because they are vertical angles.
- Therefore, $\triangle EKG \cong \triangle PKM$ by ASA (Angle-Side-Angle).
- By CPCTC, $\overline{EG} \cong \overline{MP}$.
3. Proof:
- Given: $\overline{RT} \parallel \overline{WQ}$.
- Given: $\angle R \cong \angle Q$.
- Since $\overline{RT} \parallel \overline{WQ}$, $\angle RTW \cong \angle QWT$ (alternate interior angles).
- $\overline{TW} \cong \overline{TW}$ by the Reflexive Property.
- Therefore, $\triangle RTW \cong \triangle QWT$ by AAS (Angle-Angle-Side).
- By CPCTC, $\overline{RW} \cong \overline{TQ}$.
4. Proof:
- Given: $\angle A \cong \angle C$.
- Given: $\angle 1 \cong \angle 2$.
- $\overline{BD} \cong \overline{BD}$ by the Reflexive Property.
- Therefore, $\triangle ABD \cong \triangle CBD$ by AAS (Angle-Angle-Side).
- By CPCTC, $\angle 3 \cong \angle 4$.
- Since $\angle 3 \cong \angle 4$, $\overline{BD}$ bisects $\angle ADC$.
5. Proof:
- Given: H is the midpoint of $\overline{GM}$, so $\overline{GH} \cong \overline{MH}$.
- Given: H is the midpoint of $\overline{EK}$, so $\overline{EH} \cong \overline{KH}$.
- $\angle GHE \cong \angle MHK$ because they are vertical angles.
- Therefore, $\triangle GHE \cong \triangle MHK$ by SAS (Side-Angle-Side).
- By CPCTC, $\overline{EG} \cong \overline{MK}$.
Parent Tip: Review the logic above to help your child master the concept of geometry triangle proof worksheet.
