Prove Altitudes of an isosceles triangle are congruent, drawn on the equal sides
Answers
Step-by-step explanation:
Construct an auxiliary line through point C bisecting ∠C. An angle has a unique angle bisector. Label the intersection with the base as D.
• m∠ACD = m∠BCD because an angle bisector forms two congruent angles which have equal measure.
• Under a reflection in , the reflection of C will be C, since C lies on the line of reflection.
• Since m∠ACD = m∠BCD and reflections preserve angle measure, the image of ∠ACD will be the same measure as ∠BCD.
• Since these angles are equal in measure, the reflection of ray  (side of the ∠) will coincide with its image  (side of the image angle).
• The reflection of  will have the same length as that of  since reflections preserve length.
• The reflection of  will have the same length as that of by substitution.
• The reflection of A is B since reflections preserve length and the segments share point C.
• The reflections of  and the reflection of since reflections map rays to rays.
• The reflection of ∠CAB will have the same measure as ∠CBA since reflections preserve angle measure. We have established that the rays forming these angles coincide under a reflection.
•  since congruent angles are angles of equal measure.
QED
Answer and Explanation:
To prove that the altitudes of the congruent legs in an isosceles triangle are congruent, we will make use of the following rules, properties, and theorems.
The angles opposite each of the congruent sides of an isosceles triangle are congruent, meaning they have the same measure.
If two triangles have two sets of corresponding angles being equal in measure, then the third corresponding angles of the two triangles also have the same measure.
If two triangles are such that two angles and the included side of one triangle have the same measures as two angles and the included side of the other triangle, then the two triangles are congruent, meaning that all of their corresponding sides and angles have equal measure.
Now, let's prove that the altitudes of the congruent legs in an isosceles triangle are congruent.
Let triangle ABC be an isosceles triangle with AC and BC being the legs that are congruent. Let AD be the altitude of side BC, and let BE be the altitude of side AC.
We want to show that AD is congruent to BE. Since triangle ABC is isosceles, and the sides of equal length are AC and BC, the first rule gives that angle CAB and angle CBA are congruent, so angle DBA and angle EAB are congruent. Also, since EB and DA are altitudes, we know that angle BDA and angle AEB are right angles, both with a measure of 90°. Thus, triangle EBA and triangle DAB have two sets of corresponding angles of equal measure, so by the second rule we have that the third corresponding angles, angle DAB and angle EBA, also have equal measure, or are congruent.
Notice that side AB is shared by both triangle EBA and triangle DAB, so the two triangles have the side AB that is congruent in both triangles, and the included angles of that side are congruent in both triangles. Therefore, triangle EBA and triangle DAB satisfy the third property, so the two triangles are congruent, meaning that their corresponding sides and angles are congruent. Since side AD and BE are corresponding sides of these two congruent triangles, we have that these two sides are congruent. This proves that the altitudes of the congruent legs of an isosceles triangles are congruent.
