how to prove
1)the measure of each angle of an equilateral triangle is 60.
2)the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
Thanks.
1)
The angles in any triangle have to add up to 180 degrees, right? 3 x 60 = 180
2)
??
2.
The vertex angle is 60o. If that is bisected, that makes two 30o angles of the vertex angle. And that makes the angle at the base of the triangle a right angle so we have two 30-60-90 degree triangles. The hypotenuse of each triangle is an original side of the original equilateral triangle and the sin 30 = base/hypotenuse. Since the angle is the same and the hypotenuse is the same length, then the length of each part of the base must be the same. I'm sure there is a more esoteric way of proving this but I think this covers the basics.
i don't know, i just need the definiton of this theorem
To prove the statements:
1) Let's consider an equilateral triangle ABC, where all three sides are equal in length. We want to prove that each angle of the triangle measures 60 degrees.
To begin, draw a line CD from vertex C to the midpoint D of side AB. Since ABC is an equilateral triangle, all sides are equal in length. Therefore, AD = BD.
Now, let's examine the triangles ADC and BDC. They share side CD (which is equal in length) and AD = BD. By the Side-Side-Side congruence criterion, we can conclude that triangle ADC is congruent to triangle BDC.
Since ADC and BDC are congruent, the corresponding angles are congruent. Therefore, angle ACD = angle BCD.
Now, let's add up the angles of triangle ABC. Angle ACB + angle BAC + angle ABC = 180 degrees (sum of angles in a triangle).
Since angle ACD and angle BCD are congruent and are angles in triangle ACB, we can represent angle ACD as "x" and angle BCD as "x". Therefore, we have:
x + x + angle ABC = 180 degrees
2x + angle ABC = 180 degrees
Since the sum of the angles in triangle ABC is 180 degrees, angle ABC must be equal to:
angle ABC = 180 degrees - 2x
We want to prove that angle ABC measures 60 degrees. Let's substitute the value of angle ABC into the equation:
180 degrees - 2x = 60 degrees
-2x = -120 degrees
x = -120 degrees / -2
x = 60 degrees
Since x represents both angle ACD and angle BCD, we can conclude that each angle of an equilateral triangle measures 60 degrees.
2) Let's consider an isosceles triangle ABC, where AB = AC. The vertex angle, angle BAC, is the angle opposite the base of the triangle.
To prove that the bisector of angle BAC is the perpendicular bisector of the base BC, we can construct an auxiliary line perpendicular to BC from vertex A. Let this line intersect BC at point D.
By construction, we know that AD is the perpendicular bisector of BC. Now, our goal is to prove that AD is also the bisector of angle BAC.
To do so, we need to show that angle BAD is congruent to angle DAC.
Since AB = AC (given that it is an isosceles triangle), and AD is perpendicular to BC (by construction), we have the following:
BD = DC (since AD is the perpendicular bisector of BC)
AB = AC (given that it is an isosceles triangle)
By the Side-Angle-Side congruence criterion, we can conclude that triangle ABD is congruent to triangle ACD.
Since triangle ABD and triangle ACD are congruent, the corresponding angles are congruent. Therefore, angle BAD is congruent to angle DAC.
As a result, we have proven that the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.