Use the image to answer the question. Which of the following statements is true about the triangles? (1 point) Responses by SSS Image with alt text: Triangle Upper J Upper K Upper N congruent to triangle Upper M Upper K Upper L by SSS by AAS Image with alt text: Triangle Upper J Upper K Upper N congruent to triangle Upper N Upper K Upper J by AAS by ASA Image with alt text: Triangle Upper J Upper K Upper N congruent to triangle Upper K Upper L Upper M by ASA by SAS Image with alt text: Triangle Upper J Upper K Upper N congruent to triangle Upper L Upper K Upper M by SAS
is an Isosceles triangle with a vertex angle I. If the measure of , then find the measure of . (1 point) Responses 15° 15° 45° 45° 90° 90° 30°
If triangle IJK is an isosceles triangle with a vertex angle I, and the measure of angle J is 45°, then the measure of angle K can be found using the fact that the angles in a triangle add up to 180°.
Since triangle IJK is isosceles, angles J and K are congruent. Therefore, if angle J measures 45°, then angle K also measures 45°.
So, the measure of angle K is 45°.
Based on the given information, the true statement about the triangles is:
Triangle JKN is congruent to triangle LKM by SAS.
Justify the last two steps of the proof. Given: ABCD is a parallelogram. Prove: ΔABC ΔCDA ABDC is a parallelogram. AB DC and BC DA AC CA ΔABC ΔCDA Given Opposite sides of a parallelogram are congruent. _____________________? _____________________? (1 point) Responses Transitive Property of ; SAS Transitive Property of Image with alt text: congruent ; SAS Reflexive Property of ; SAS Reflexive Property of Image with alt text: congruent ; SAS Transitive Property of ; SSS Transitive Property of Image with alt text: congruent ; SSS Reflexive Property of ; SSS Reflexive Property of Image with alt text: congruent ; SSS Skip to navigation
The last two steps of the proof can be justified as follows:
1. Opposite sides of a parallelogram are congruent: This is a property of parallelograms, and it states that in a parallelogram, the opposite sides are equal in length.
2. ΔABC ≅ ΔCDA (Triangle congruence): This is justified by the Side-Angle-Side (SAS) congruence criterion. The sides AB and CD are congruent (property of parallelograms), the side BC is congruent to the side DA (property of parallelograms), and the included angle ∠ABC is congruent to the included angle ∠CDA (corresponding angles of congruent triangles). Therefore, by the SAS congruence criterion, we can conclude that ΔABC is congruent to ΔCDA.
Therefore, based on the congruence of ΔABC and ΔCDA, we can conclude that ABDC is a parallelogram.
Justify the last two steps of the proof. Given: ABCD is a parallelogram. Prove: ΔABC ΔCDA ABDC is a parallelogram. AB DC and BC DA AC CA ΔABC ΔCDA Given Opposite sides of a parallelogram are congruent. _____________________? _____________________? (1 point) Responses
Transitive Property of ; SAS
Reflexive Property of ; SAS
Transitive Property of ; SSS
Reflexive Property of ; SSS
The last two steps of the proof can be justified as follows:
1. Opposite sides of a parallelogram are congruent: This is a property of parallelograms, which states that in a parallelogram, the opposite sides are equal in length.
2. ΔABC ≅ ΔCDA (Triangle congruence using SAS): This is justified by the Side-Angle-Side (SAS) congruence criterion. We have AB congruent to CD (opposite sides of the parallelogram) and BC congruent to DA (opposite sides of the parallelogram). Additionally, we have ∠ABC congruent to ∠CDA (corresponding angles of congruent triangles). Therefore, based on the SAS congruence criterion, we can conclude that ΔABC is congruent to ΔCDA.
Therefore, based on the congruence of ΔABC and ΔCDA, we can conclude that ABDC is a parallelogram.
11 of 1611 of 16 Items Question What are the missing reasons in the two-column proof? Given: bisects and bisects Prove: A two column proof is shown. In the statements column there are 6 statements. 1 says modifying above with right arrow upper M upper O bisects angle upper P upper M upper N. 2 says angle upper P upper M upper O congruent to angle upper N upper M upper O. 3 says modifying above with bar upper M upper O congruent to modifying above with bar upper M upper O. 4 says modifying above with right arrow upper O upper M bisects angle upper P upper O upper N. 5 says angle upper P upper O upper M congruent to angle upper N upper O upper M. 6 says triangle upper P upper M upper O congruent to triangle upper N upper M upper O. In the reasons columns there are the numbers 1 through 6 but each reason is blank with a question mark. The diagram is also shown. Line upper M upper O is horizontal across the middle of the diagram. Point upper N is at the top of the diagram and two lines extend from it down to line upper M upper O forming a triangle. Another point upper P is at the bottom of the diagram and two lines extend from it up to line upper M upper O forming another triangle. The two triangles appear congruent and share two vertices on line upper M upper O. (6 points) BoldItalicUnderlineBullet listNumbered listUndoRedo Skip to navigation
Based on the given information and the diagram, we can fill in the missing reasons in the two-column proof as follows:
1. ∠PMO is bisected by MO (Given)
2. ∠PMO ≅ ∠NMO (Definition of angle bisector)
3. MO ≅ MO (Reflexive Property)
4. ∠POM is bisected by OM (Given)
5. ∠POM ≅ ∠NOM (Definition of angle bisector)
6. ΔPMO ≅ ΔNMO (ASA congruence criterion)
Therefore, the missing reasons in the two-column proof are:
1. Definition of angle bisector
2. Reflexive Property
3. Definition of angle bisector
4. ASA congruence criterion