### Educational Explanation of Triangle Congruence#### Given Problem:We have two triangles, \( \triangle DEF \) and \( \triangle PQR \), displayed in a diagram. Each triangle has a right angle, marked at vertices \( E \) and \( Q \), respectively. The sides are marked such that \( \overline{DF} \cong \overline{PR} \) and \( \overline{DE} \cong \overline{QR} \).#### Two-Column Proof:| Statement | Reason ||----------------------------------------|------------------------------------|| \( \angle E \) and \( \angle Q \) are right angles | Given || \( \triangle DEF \) and \( \triangle PQR \) are right triangles | Definition of right triangle || \( \overline{DF} \cong \overline{PR} \) | Given || \( \overline{DE} \cong \overline{QR} \) | Given || \( \triangle DEF \cong \triangle PQR \) | ? |#### Missing Reason:The missing reason is the **Hypotenuse-Leg Triangle Congruence Theorem**.This theorem states that two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of the other triangle.#### Conclusion:With the right angle, congruent hypotenuses \( \overline{DF} \) and \( \overline{PR} \), and congruent legs \( \overline{DE} \) and \( \overline{QR} \), \( \triangle DEF \cong \triangle PQR \) by the Hypotenuse-Leg Triangle Congruence Theorem.For more information on triangle congruence theorems and proofs, please refer to relevant educational resources.

ADEF and APQR are shown below: VA P R F A teacher writes the following 2-column proof on the board. Statement Reason ZE and 2Q are right angles Given A DEF and APQR are right triangles Definition of right triangle DF = PR Given DE = QR Given A DEF = A PQR ? What is the missing reason? Side-Side-Side Triangle Congruence Theorem Hypotenuse-Leg Triangle Congruence Theorem Angle-Side-Angle Triangle Congruence Theorem

ADEF and APQR are shown below: VA P R F A teacher writes the following 2-column proof on the board. Statement Reason ZE and 2Q are right angles Given A DEF and APQR are right triangles Definition of right triangle DF = PR Given DE = QR Given A DEF = A PQR ? What is the missing reason? Side-Side-Side Triangle Congruence Theorem Hypotenuse-Leg Triangle Congruence Theorem Angle-Side-Angle Triangle Congruence Theorem

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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