It is given that , and . Show that is parallel to . [Hint: is a transversal for these two lines. Are there any equal alternate angles?]

Understand the Question We are given two pairs of equal segments: and . The line segments and intersect at , which gives a pair of vertically opposite angles: . Using the SAS (Side-Angle-Side) congruence criterion, we prove . By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the alternate interior angles and are equal, proving that . Step 1 · Prove In and : (Given) (Vertically opposite angles) (Given) Therefore, by the SAS congruence criterion: Step 2 · Show Since , their corresponding angles are equal by CPCTC: For line segments and intersected by the transversal , and form a pair of alternate interior angles. Since alternate interior angles are equal: Answer > Hence proved, . Common Mistakes Incorrect Triangle Correspondence: Writing instead of . The correspondence must match , , and . Missing Angle Justification: Forgetting to explicitly state that because they are vertically opposite angles.

Geometric Twins | FIO

Question 16

It is given that $OB = OC$ , and $OA = OD$ . Show that $AB$ is parallel to $CD$ .
[Hint: $AD$ is a transversal for these two lines. Are there any equal alternate angles?]

Solution

Understand the Question

We are given two pairs of equal segments: $OB = OC$ and $OA = OD$ .
The line segments $AD$ and $BC$ intersect at $O$ , which gives a pair of vertically opposite angles: $\angle AOB = \angle DOC$ .
Using the SAS (Side-Angle-Side) congruence criterion, we prove $\triangle AOB \cong \triangle DOC$ .
By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the alternate interior angles $\angle OAB$ and $\angle ODC$ are equal, proving that $AB \parallel CD$ .

Step 1 · Prove $\triangle AOB \cong \triangle DOC$

Diagram 1

In $\triangle AOB$ and $\triangle DOC$ :

$OA = OD$ (Given)
$\angle AOB = \angle DOC$ (Vertically opposite angles)
$OB = OC$ (Given)

Therefore, by the SAS congruence criterion: $\triangle AOB \cong \triangle DOC$

Step 2 · Show $AB \parallel CD$

Since $\triangle AOB \cong \triangle DOC$ , their corresponding angles are equal by CPCTC: $\angle OAB = \angle ODC$

For line segments $AB$ and $CD$ intersected by the transversal $AD$ , $\angle OAB$ and $\angle ODC$ form a pair of alternate interior angles.

Since alternate interior angles are equal: $AB \parallel CD$

Answer

Hence proved, $AB \parallel CD$ .

Common Mistakes

Incorrect Triangle Correspondence: Writing $\triangle AOB \cong \triangle CDO$ instead of $\triangle AOB \cong \triangle DOC$ . The correspondence must match $A \leftrightarrow D$ , $O \leftrightarrow O$ , and $B \leftrightarrow C$ .
Missing Angle Justification: Forgetting to explicitly state that $\angle AOB = \angle DOC$ because they are vertically opposite angles.

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Q16

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[Hint: $AD$ is a transversal for these two lines. Are there any equal alternate angles?]

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