Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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**Understanding Rigid Motions in Geometry** In the diagram below, a sequence of rigid motions maps the quadrilateral \(ABCD\) onto \(JKLM\). ![Diagram of Rigid Motions](link-to-diagram.png) The diagram features two quadrilaterals on a coordinate plane: 1. Quadrilateral \(ABCD\) positioned in the first quadrant. 2. Quadrilateral \(JKLM\) positioned in the third quadrant. ### Axes: - **X-axis**: Horizontal axis of the coordinate plane - **Y-axis**: Vertical axis of the coordinate plane ### Vertices: - **\(ABCD\)**: - \(A\) is positioned at top right. - \(B\) is located slightly below and to the left of \(A\). - \(C\) is below \(B\). - \(D\) is to the right of \(C\). - **\(JKLM\)**: - \(J\) is positioned to the top right. - \(K\) is to the left of \(J\). - \(L\) is below \(K\). - \(M\) is right of \(L\). ### Given Angles: - \(\angle A = 82^\circ\) - \(\angle B = 104^\circ\) - \(\angle L = 121^\circ\) ### Unknown Angle: - \(\angle M\) The task is to determine the measure of \(\angle M\). Given the angles and the nature of rigid motions, which preserve angles and relative side lengths: - Rigid motions include translations, rotations, and reflections. - The sum of angles in any quadrilateral is \(360^\circ\). ### Calculation: Sum of angles in \(JKLM\): \[ \angle J + \angle K + \angle L + \angle M = 360^\circ \] Given: \[ \angle J \equiv \angle A = 82^\circ \] \[ \angle K \equiv \angle B = 104^\circ \] \[ \angle L = 121^\circ \] To find \(\angle M\): \[ 82^\circ + 104^\circ + 121^\circ + \angle M = 360^\circ \] \[ 307^\circ + \angle M = 360^\circ \] \[ \angle M = 360
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Transcribed Image Text:**Understanding Rigid Motions in Geometry** In the diagram below, a sequence of rigid motions maps the quadrilateral \(ABCD\) onto \(JKLM\). ![Diagram of Rigid Motions](link-to-diagram.png) The diagram features two quadrilaterals on a coordinate plane: 1. Quadrilateral \(ABCD\) positioned in the first quadrant. 2. Quadrilateral \(JKLM\) positioned in the third quadrant. ### Axes: - **X-axis**: Horizontal axis of the coordinate plane - **Y-axis**: Vertical axis of the coordinate plane ### Vertices: - **\(ABCD\)**: - \(A\) is positioned at top right. - \(B\) is located slightly below and to the left of \(A\). - \(C\) is below \(B\). - \(D\) is to the right of \(C\). - **\(JKLM\)**: - \(J\) is positioned to the top right. - \(K\) is to the left of \(J\). - \(L\) is below \(K\). - \(M\) is right of \(L\). ### Given Angles: - \(\angle A = 82^\circ\) - \(\angle B = 104^\circ\) - \(\angle L = 121^\circ\) ### Unknown Angle: - \(\angle M\) The task is to determine the measure of \(\angle M\). Given the angles and the nature of rigid motions, which preserve angles and relative side lengths: - Rigid motions include translations, rotations, and reflections. - The sum of angles in any quadrilateral is \(360^\circ\). ### Calculation: Sum of angles in \(JKLM\): \[ \angle J + \angle K + \angle L + \angle M = 360^\circ \] Given: \[ \angle J \equiv \angle A = 82^\circ \] \[ \angle K \equiv \angle B = 104^\circ \] \[ \angle L = 121^\circ \] To find \(\angle M\): \[ 82^\circ + 104^\circ + 121^\circ + \angle M = 360^\circ \] \[ 307^\circ + \angle M = 360^\circ \] \[ \angle M = 360
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