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If the bisector of the vertical angle of the triangle bisects the base prove that the triangle is isosceles?
Wiki User
∙ 11y ago
show that the bisector of the vertical angle of an isosceles triangle bisects the base
Wiki User
∙ 11y ago
Anonymous
Given:
In ∆ABC ,
AD bisects ∠BAC, & BD= CD
To Prove:
AB=AC
Construction:
Produce AD to E such that AD=DE & then join E to C.
Proof:
In ∆ADB & ∆EDC
AD= ED ( by construction)
∠ADB= ∠EDC. (vertically opposite angles (
BD= CD (given)
∆ADB congruent ∆EDC (by SAS)
Hence, ∠BAD=∠CED......(1) (CPCT)
∠BAD=∠CAD......(2). (given)
From eq.1 &2
∠CED =∠CAD......(3)
AB=CE (CPCT).......(4)
From eq 3 as proved that
∠CED=∠CAD
So we can say CA=CE......(5)
[SIDES OPPOSITE TO EQUAL ANGLES ARE EQUAL]
Hence, from eq 4 & 5
AB = AC
HENCE THE ∆ IS ISOSCELES..
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Hope this will help you....
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If you extend a ray that bisects an angle of a triangle so that it passes through the opposite side will it bisect the opposite side?
Only in an equilateral triangle will bisectors of the three angles bisect the opposite sides. In an isosceles triangle, only the bisector of the one different angle will bisect the opposite side (between the identical angles).
Angle bisector of angleA of triangleABC is perpendicular to BC prove it is isosceles?
Let D represent the point on BC where the bisector of A intersects BC. Because AD bisects angle A, angle BAD is congruent to CAD. Because AD is perpendicular to BC, angle ADB is congruent to ADC (both are right angles). The line segment is congruent to itself. By angle-side-angle (ASA), we know that triangle ADB is congruent to triangle ADC. Therefore line segment AB is congruent to AC, so triangle ABC is isosceles.
Do the medians of a triangle bisect the internal angles?
Not always. 1. The median to the base of an isosceles triangle bisects the vertex angle. 2. When the triangle is an equilateral triangle, then the medians bisect the interior angles of the triangle.
What Angle bisector of a triangle?
An angle is formed by two rays with a common endpoint. The angle bisector is a ray or line segment that bisects the angle, creating two congruent angles. To construct an angle bisector you need a compass and straightedge. Bisectors are very important in identifying corresponding parts of similar triangles and in solving proofs.
How would you construct an isosceles triangle if only given the vertex angle and the radius of the circumscribed circle?
You have an isosceles triangle, and a circle that is drawn around it. You know the vertex angle of the isosceles triangle, and you know the radius of the circle. If you use a compass and draw the circle according to its radius, you can begin your construction. First, draw a bisecting cord vertically down the middle. This bisects the circle, and it will also bisect your isosceles triangle. At the top of this cord will be the vertex of your isosceles triangle. Now is the time to work with the angle of the vertex. Take the given angle and divide it in two. Then take that resulting angle and, using your protractor, mark the angle from the point at the top of the cord you drew. Then draw in a line segment from the "vertex point" and extend it until it intersects the circle. This new cord represents one side of the isosceles triangle you wished to construct. Repeat the process on the other side of the vertical line you bisected the circle with. Lastly, draw in a line segment between the points where the two sides of your triangle intersect the circle, and that will be the base of your isosceles triangle.
What is a segment that bisects one the angles of the triangle?
A bisector
Can an altitude be a perpendicular bisector?
Yes - the altitude of an equilateral triangle is perpendicular to the side chosen as the base and bisects that side and the opposite angle. Also, the altitude of an isosceles triangle when measured from the third side (the side that is not equal to the other two sides) is a perpendicular bisector of the base and also bisects the opposite angle.
Perpendicular bisector of a triangle?
Is a line that bisects a side of a triangle and is perpendicular to that side.
How do you determine that an isosceles triangle is an isosceles right triangle?
If the non-right angles are 45 degrees each. or If the sides adjacent to the right angle are equal. There are other properties that may be used instead. For example, the perpendicualr bisector of the hypotenuse bisects the right angle of the triangle.
How you can draw a altitude with isosceles triangle?
The altitude line is perpendicular to the base and bisects the apex of the isosceles triangle.
How does the perpendicular bisector affect the vertex angle of an isoceles triangle?
The perpendicular bisector bisects the angle at the vertex.
How do the Triangle-Angle Bisector Theorem and the Angle Bisector Theorem differ?
They are the same concept, one for the angle and 1 for triangle.Definition of a triangle angle bisector is a line segment that bisects one of the vertex angles of a triangle.Definition of an angle bisector is a ray or line segment that bisects the angle, creating two congruent angles.
A perpendicular line or segment that bisects one side of a triangle is called the?
Perpendicular bisector.
Why are the median angle bisector and altitude from the vertex angle of an isoceles triangle all the same?
It's in the definition of an angle bisector: An angle is formed by two rays with a common endpoint. The angle bisector is a ray or line segment that bisects the angle, creating two congruent angles.and an isoceles triangle:it is a triangle with (at least) two equal sides. This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles.
If you extend a ray that bisects an angle of a triangle so that it passes through the opposite side will it bisect the opposite side?
Only in an equilateral triangle will bisectors of the three angles bisect the opposite sides. In an isosceles triangle, only the bisector of the one different angle will bisect the opposite side (between the identical angles).
What bisects an angle?
The Angle Bisector Theorem states that given triangle and angle bisector AD, where D is on side BC, then . Likewise, the converse is also true. Not sure if this is what you want?
What is the difference between and angle bisector and a perpendicular bisector?
An angle bisector bisects an angle. A perpendicular bisector bisects a side.
