given: $m$ is the midpoint of $\overline{pk}$

$\overline{pk} \perp \overline{mb}$

prove: $\delta pkb$ is isosceles

midpoints divide a segment into two congruent segments, so $\overline{pm} \cong \overline{km}$. since $\overline{pk} \perp \overline{mb}$ and perpendicular lines intersect at right angles, $\angle pmb$ and $\angle kmb$ are right angles. right angles are congruent, so $\angle pmb \cong \angle kmb$. the triangles share $\overline{mb}$, and the reflexive property justifies that $\overline{mb} \cong \overline{mb}$. therefore, $\delta pmb \cong \delta kmb$ by the sas congruence theorem. thus, $\overline{bp} \cong \overline{bk}$ because _______. finally, $\delta pkb$ is isosceles because it has two congruent sides.

corresponding parts of congruent triangles are congruent
base angles of isosceles triangles are congruent
of the definition of congruent segments
of the definition of a right triangle

Question

given: $m$ is the midpoint of $\overline{pk}$

$\overline{pk} \perp \overline{mb}$

prove: $\delta pkb$ is isosceles

midpoints divide a segment into two congruent segments, so $\overline{pm} \cong \overline{km}$. since $\overline{pk} \perp \overline{mb}$ and perpendicular lines intersect at right angles, $\angle pmb$ and $\angle kmb$ are right angles. right angles are congruent, so $\angle pmb \cong \angle kmb$. the triangles share $\overline{mb}$, and the reflexive property justifies that $\overline{mb} \cong \overline{mb}$. therefore, $\delta pmb \cong \delta kmb$ by the sas congruence theorem. thus, $\overline{bp} \cong \overline{bk}$ because _______. finally, $\delta pkb$ is isosceles because it has two congruent sides.

corresponding parts of congruent triangles are congruent
base angles of isosceles triangles are congruent
of the definition of congruent segments
of the definition of a right triangle

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Triangle Congruence Theorems", "SAS Congruence Postulate" ], "new_concepts": [ "CPCTC" ], "current_concepts": [ "Triangle Congruence Theorems", "SAS Congruence Postulate", "CPCTC" ] } ## Analyze the given proof context The proof establishes that $\Delta PMB \cong \Delta KMB$ using the **SAS Congruence Postulate**. ## Identify the missing reason The next statement in the proof is: "Thus, $\overline{BP} \cong \overline{BK}$ because _________." Since $\overline{BP}$ and $\overline{BK}$ are corresponding sides of the congruent triangles $\Delta PMB$ and $\Delta KMB$, they are congruent by the principle that corresponding parts of congruent triangles are congruent (CPCTC). ## Match with the given options The first option states: "corresponding parts of congruent triangles are congruent". This matches the CPCTC principle perfectly.

corresponding parts of congruent triangles are congruent

base angles of isosceles triangles are congruent

of the definition of congruent segments

of the definition of a right triangle

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Congruence Theorems" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze the given proof context

Identify the missing reason

Match with the given options

Answer: