Asked by Ian
Given: A, B, C, DABCD is a parallelogram and triangle, A, B, C, \cong, triangle, D, C, B, .△ABC≅△DCB.
Prove: start overline, A, B, end overline, is perpendicular to, start overline, B, C, end overline
AB
⊥
BC
.
In the table below, the first step of the proof is completed for you. To complete each following step, first use the dropdown menu in the "Statement" column to select the format of a statement. Once you select a format, a statement will appear with input boxes and/or dropdowns to complete, and a dropdown menu will appear in the reason column. Once you complete these inputs and dropdowns, navigate back to the first column and press the "try" button. If your statement and reason are valid, a new row will appear below. This process will continue until you have filled in a statement and reason for what you were asked to prove, and then the "try" button will cause the answer area to appear with a submit button. The diagram associated with this problem is located below the proof table, and is available as a tactile printout. There may be a clarifying note below the diagram.
Step Statement Reason
1
A, B, C, DABCD is a parallelogram
triangle, A, B, C, \cong, triangle, D, C, B△ABC≅△DCB
Given. What are the remaining steps, statements, and reasons?
All Answers 1
Answered by
GPT-5 mini
AI
1) A, B, C, D; ABCD is a parallelogram. △ABC ≅ △DCB. — Given.
2) AC = BD. — Corresponding parts of congruent triangles are congruent (CPCTC).
3) Let E be the intersection of diagonals AC and BD. — Definition (diagonals intersect).
4) AE = CE and BE = DE. — Diagonals of a parallelogram bisect each other.
5) AE = BE. — From AC = BD, divide both sides by 2 (AC = 2·AE, BD = 2·BE).
6) Hence AE = BE = CE = DE. — From steps 4 and 5 (transitivity of equality).
7) A, B, C, D are concyclic (all four vertices are equidistant from E). — Definition of a circle / concyclic points.
8) Opposite angles of a cyclic quadrilateral are supplementary, and opposite angles of a parallelogram are equal; therefore each pair of opposite angles must be 180° and equal, so each is 90°. In particular ∠ABC = 90°. — Properties of cyclic quadrilaterals and parallelograms.
9) AB ⟂ BC. — Definition of perpendicular lines (angle of 90°).
Thus AB is perpendicular to BC.
2) AC = BD. — Corresponding parts of congruent triangles are congruent (CPCTC).
3) Let E be the intersection of diagonals AC and BD. — Definition (diagonals intersect).
4) AE = CE and BE = DE. — Diagonals of a parallelogram bisect each other.
5) AE = BE. — From AC = BD, divide both sides by 2 (AC = 2·AE, BD = 2·BE).
6) Hence AE = BE = CE = DE. — From steps 4 and 5 (transitivity of equality).
7) A, B, C, D are concyclic (all four vertices are equidistant from E). — Definition of a circle / concyclic points.
8) Opposite angles of a cyclic quadrilateral are supplementary, and opposite angles of a parallelogram are equal; therefore each pair of opposite angles must be 180° and equal, so each is 90°. In particular ∠ABC = 90°. — Properties of cyclic quadrilaterals and parallelograms.
9) AB ⟂ BC. — Definition of perpendicular lines (angle of 90°).
Thus AB is perpendicular to BC.
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