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Statements 1. KJ MN and LKJL LMNL 2. LKLJ LMLN Reasons 1. 2. 3. 3.

Statements 1. KJ MN and LKJL LMNL 2. LKLJ LMLN Reasons 1. 2. 3. 3.

Elementary Geometry For College Students, 7e
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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### Educational Material: Proving Triangle Congruence **Given:** \[ \overline{KJ} \cong \overline{MN} \text{ and } \angle KJL \cong \angle MNL \] **To Prove:** \[ \triangle JKL \cong \triangle NML \] #### Statement and Reason Table: To prove the triangles congruent, we need to construct a logical argument with statements followed by reasons. | Statements | Reasons | |------------|---------| | 1. \( \overline{KJ} \cong \overline{MN} \) and \( \angle KJL \cong \angle MNL \) | 1. Given | | 2. \( \angle KLJ \cong \angle MLN \) | 2. Vertical angles are congruent. | | 3. \( \triangle JKL \cong \triangle NML \) | 3. ASA (Angle-Side-Angle) Congruence Postulate | The given image contains a geometric diagram illustrating two triangles, \( \triangle JKL \) and \( \triangle NML \), with the following congruences: - \( \overline{KJ} \) is congruent to \( \overline{MN} \) - \( \angle KJL \) is congruent to \( \angle MNL \) Additionally, it depicts vertical angles \( \angle KLJ \) and \( \angle MLN \) being congruent, which is a key step in proving the congruence of the two triangles using the Angle-Side-Angle (ASA) postulate. ### Explanation of Diagrams Above the given statements and reasons table, there's a diagram of two triangles. The triangles share common angular relationships due to their arrangement with intersecting lines. Points J, K, L, M, and N are labeled, and congruent segments and angles are marked respectively. **Steps for the Proof:** 1. Identify the given congruent segment and angle. 2. Recognize the pair of vertical angles between the triangles. 3. Conclude the triangles are congruent using ASA (Angle-Side-Angle) postulate. This proof strategy leverages the given information and geometric properties to establish the congruence of the triangles systematically.
Transcribed Image Text:### Educational Material: Proving Triangle Congruence **Given:** \[ \overline{KJ} \cong \overline{MN} \text{ and } \angle KJL \cong \angle MNL \] **To Prove:** \[ \triangle JKL \cong \triangle NML \] #### Statement and Reason Table: To prove the triangles congruent, we need to construct a logical argument with statements followed by reasons. | Statements | Reasons | |------------|---------| | 1. \( \overline{KJ} \cong \overline{MN} \) and \( \angle KJL \cong \angle MNL \) | 1. Given | | 2. \( \angle KLJ \cong \angle MLN \) | 2. Vertical angles are congruent. | | 3. \( \triangle JKL \cong \triangle NML \) | 3. ASA (Angle-Side-Angle) Congruence Postulate | The given image contains a geometric diagram illustrating two triangles, \( \triangle JKL \) and \( \triangle NML \), with the following congruences: - \( \overline{KJ} \) is congruent to \( \overline{MN} \) - \( \angle KJL \) is congruent to \( \angle MNL \) Additionally, it depicts vertical angles \( \angle KLJ \) and \( \angle MLN \) being congruent, which is a key step in proving the congruence of the two triangles using the Angle-Side-Angle (ASA) postulate. ### Explanation of Diagrams Above the given statements and reasons table, there's a diagram of two triangles. The triangles share common angular relationships due to their arrangement with intersecting lines. Points J, K, L, M, and N are labeled, and congruent segments and angles are marked respectively. **Steps for the Proof:** 1. Identify the given congruent segment and angle. 2. Recognize the pair of vertical angles between the triangles. 3. Conclude the triangles are congruent using ASA (Angle-Side-Angle) postulate. This proof strategy leverages the given information and geometric properties to establish the congruence of the triangles systematically.
In the diagram above, we observe two triangles, KJL and MNL, positioned such that they share a common vertex, L. Here is a detailed explanation of the diagram: 1. **Triangles Structure**: - Triangle KJL on the left. - Triangle MNL on the right. 2. **Common Vertex**: The point L is the common vertex where the two triangles intersect. 3. **Angles**: - Angle KJL in triangle KJL. - Angle MNL in triangle MNL. 4. **Equal Angles**: - Both angles at vertices J and N are equal. - The diagram uses a single arc inside these angles to denote their equality. 5. **Equal Sides**: - The sides opposite to the vertex L (KJ and NM) have a single tick mark, indicating these sides are of equal length. .*; In summary, the diagram is illustrating two triangles (KJL and MNL) that share a vertex (L) with equal corresponding angles (at vertices J and N) and sides (KJ and NM). This kind of configuration could be a representation of angle-side-angle (ASA) postulate in geometry, showcasing how two triangles can be congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle.
Transcribed Image Text:In the diagram above, we observe two triangles, KJL and MNL, positioned such that they share a common vertex, L. Here is a detailed explanation of the diagram: 1. **Triangles Structure**: - Triangle KJL on the left. - Triangle MNL on the right. 2. **Common Vertex**: The point L is the common vertex where the two triangles intersect. 3. **Angles**: - Angle KJL in triangle KJL. - Angle MNL in triangle MNL. 4. **Equal Angles**: - Both angles at vertices J and N are equal. - The diagram uses a single arc inside these angles to denote their equality. 5. **Equal Sides**: - The sides opposite to the vertex L (KJ and NM) have a single tick mark, indicating these sides are of equal length. .*; In summary, the diagram is illustrating two triangles (KJL and MNL) that share a vertex (L) with equal corresponding angles (at vertices J and N) and sides (KJ and NM). This kind of configuration could be a representation of angle-side-angle (ASA) postulate in geometry, showcasing how two triangles can be congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle.
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