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Q1.

Expert-verified
Found in: Page 130

Geometry

Book edition Student Edition
Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen
Pages 227 pages
ISBN 9780395977279

Copy and complete the proof.1. Given: $\mathbf{\angle }\mathbit{P}\mathbf{\cong }\mathbf{\angle }\mathbit{S}\mathbf{;}\mathbf{\text{}}\mathbit{O}$ is the midpoint of $\overline{\mathbf{P}\mathbf{S}}$. Prove: is the midpoint of $\overline{\mathbf{R}\mathbf{Q}}$Proof

 Statement Reason 1. $\angle P\cong \angle S$ Given 2. $O$ is the midpoint of $\overline{PS}$ Given 3. $\overline{PO}\cong \overline{SO}$ As is the midpoint of and midpoint is a point on a line segment that divides it into two congruent line segments 4. $\angle POQ\cong \angle SOR$ Vertically Opposite angles 5. $\Delta POQ\cong \Delta SOR$ ASA congruence criteria 6. $\overline{QO}\cong \overline{RO}$ Corresponding parts of congruent triangles are equal 7. $O$ is the midpoint of $\overline{RQ}$ As and midpoint is a point on a line segment that divides it into two congruent line segments
See the step by step solution

Step 1 - Fill the blanks of first and second reason

Since, first and second statement of two column proof are given in the question. So, there reason must be “Given”.

Step 2 - Fill third blank

Since, midpoint is a point on a line segment that divides it into two congruent line segments

Thus, $O$ divides line segment $\overline{PS}$ in two congruent line segments

Hence, reason of third statement must be

“As $O$ is the midpoint of $\overline{PS}$ and midpoint is a point on a line segment that divides it into two congruent line segments”

Step 3 - Fill fourth blank

Observe from figure, $\angle POQ$ and $\angle SOR$ are vertically opposite angles and such angles are congruent

So reason for $\angle POQ\cong \angle SOR$ will be “Vertically opposite angles”

Step 4 - Fill fifth blank

According to ASA congruence criteria, two triangles are said to be congruent if two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle

Using statement 1, 3, and 4 $\Delta POQ\cong \Delta SOR$

So its reason will be “ASA Congruence Criteria”

Step 5 - Fill sixth blank

Since, corresponding parts of congruent triangles are equal

So, $\overline{QO}\cong \overline{RO}$ because of same reason

Step 6 - Fill seventh blank

As $\overline{QO}\cong \overline{RO}$ and midpoint is a point on a line segment that divides it into two congruent line segments

Thus, it the reason for the statement “$O$ is the midpoint of $\overline{RQ}$

Hence, complete proof will be

 Statement Reason 1. $\angle P\cong \angle S$ Given 2. is the midpoint of $\overline{PS}$ Given 3. $\overline{PO}\cong \overline{SO}$ As $O$ is the midpoint of $\overline{PS}$ and midpoint is a point on a line segment that divides it into two congruent line segments 4. $\angle POQ\cong \angle SOR$ Vertically Opposite angles 5. $\Delta POQ\cong \Delta SOR$ ASA congruence criteria 6. $\overline{QO}\cong \overline{RO}$ Corresponding parts of congruent triangles are equal 7. is the midpoint of As$\overline{QO}\cong \overline{RO}$ and midpoint is a point on a line segment that divides it into two congruent line segments