The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.In terms of ratios, the sine of an angle is defined, in a right angled triangle, as the ratio of lengths of the opposite side to the hypotenuse. Show
A triangle that has two sides of the same measure and the third side with a different measure is known as an isosceles triangle. The isosceles triangle theorem in math states that in an isosceles triangle, the angles opposite to the equal sides are also equal in measurement. We will be learning about the isosceles triangle theorem and its converse in this article. What is Isosceles Triangle Theorem?Isosceles triangle theorem states that if two sides of a triangle are congruent, then the angles opposite to the congruent sides are also congruent. To understand the isosceles triangle theorem, we will be using the properties of an isosceles triangle for the proof as discussed below. Isosceles Triangle Theorem ProofLet's draw an isosceles triangle with two equal sides as shown in the figure below.  Given: ∆ABC is an isosceles triangle with AB = AC. Proof: We know, that the altitude of an isosceles triangle from the vertex is the perpendicular bisector of the third side. Thus, we can conclude that, ∠ADB = ∠ADC = 90º ----------- (1) BD = DC ---------- (2) Consider ∆ADB and ∆ADC AB = AC [Given] AD = AD [common side] BD = DC [From equation (2)] Thus, by SSS congruence we can say that, ∆ADB ≅ ∆ADC By CPCT, ∠B = ∠C. Hence, we have proved that if two sides of a triangle are congruent, then the angles opposite to the congruent sides are equal. Isosceles Triangle Theorem ConverseThe converse of isosceles triangle theorem states that if two angles of a triangle are congruent, then the sides opposite to the congruent angles are equal. This is exactly the reverse of the theorem we discussed above. We will be using the properties of the isosceles triangle to prove the converse as discussed below. Converse of Isosceles Triangle Theorem ProofLet's draw a triangle with two congruent angles as shown in the figure below with the markings as indicated. Given: ∆ABC with ∠B = ∠C. Proof: We know that the altitude of a triangle is always at a right angle with the side on which it is dropped. Hence, ∠ADB = ∠ADC = 90º ----------- (1) Consider ∆ADB and ∆ADC, ∠B = ∠C [Given] AD = AD [common side] ∠ADB = ∠ADC = 90º [From equation (1)] Thus, by AAS congruence we can say that, ∆ADB ≅ ∆ADC By CPCT, AB = AC Hence we have proved that, if two angles of a triangle are congruent, then the sides opposite to the congruent angles are equal. Related ArticlesCheck these articles related to the concept of the isosceles triangle theorem.
go to slidego to slide
Want to build a strong foundation in Math? Go beyond memorizing formulas and understand the ‘why’ behind them. Experience Cuemath and get started. Book a Free Trial Class
FAQs on Isosceles Triangle TheoremIsosceles triangle theorem states that, if two sides of an isosceles triangle are equal then the angles opposite to the equal sides will also have the same measure. How to Prove Isosceles Triangle Theorem?Isosceles triangle theorem can be proved by using the congruence properties and properties of an isosceles triangle. An isosceles triangle can be drawn, followed by constructing its altitude. The two triangles now formed with altitude as its common side can be proved congruent by SSS congruence followed by proving the angles opposite to the equal sides to be equal by CPCT. What is the Converse of Isosceles Triangle Theorem?The converse of isosceles triangle theorem states that, if two angles of a triangle are equal, then the sides opposite to the equal angles of a triangle are of the same measure. How to Prove the Converse of the Isosceles Triangle Theorem?The converse of the isosceles triangle theorem can be proved by using the congruence properties and properties of an isosceles triangle. An isosceles triangle can be drawn, followed by constructing its altitude. The two triangles now formed with altitude as its common side can be proved congruent by AAS congruence followed by proving the sides opposite to the equal angles to be equal by CPCT. How to find Angles using Isosceles Triangle Theorem?The angles of an isosceles triangle add up to 180º according to the angle sum property of a triangle. The angles opposite to the equal sides of an isosceles triangle are considered to be an unknown variable 'x'. Now, if the measure of the third (unequal) angle is given, then the three angles can be added to equate it to 180º to find the value of x that gives all the angles of a triangle. For example: Let the unequal angle of an isosceles triangle be 50º. The other angles can be considered as x each as they are equal. By using the angle sum property, 50º + x + x = 180° 2x = 180º - 50º 2x = 130º x = 65º Thus, the angles of the isosceles triangle are 65º, 65º, and 50º.
Isosceles triangles have equal legs (that's what the word "isosceles" means). Yippee for them, but what do we know about their base angles? How do we know those are equal, too? We reach into our geometer's toolbox and take out the Isosceles Triangle Theorem. No need to plug it in or recharge its batteries -- it's right there, in your head! Isosceles TriangleHere we have on display the majestic isosceles triangle, △DUK. You can draw one yourself, using △DUK as a model. Hash marks show sides ∠DU ≅ ∠DK, which is your tip-off that you have an isosceles triangle. If these two sides, called legs, are equal, then this is an isosceles triangle. What else have you got? Properties of an Isosceles TriangleLet's use △DUK to explore the parts:
Isosceles Triangle TheoremKnowing the triangle's parts, here is the challenge: how do we prove that the base angles are congruent? That is the heart of the Isosceles Triangle Theorem, which is built as a conditional (if, then) statement:
The Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent. To mathematically prove this, we need to introduce a median line, a line constructed from an interior angle to the midpoint of the opposite side. We find Point C on base UK and construct line segment DC: There! That's just DUCKy! Look at the two triangles formed by the median. We are given:
We just showed that the three sides of △DUC are congruent to △DCK, which means you have the Side Side Side Postulate, which gives congruence. So if the two triangles are congruent, then corresponding parts of congruent triangles are congruent (CPCTC), which means … Converse of the Isosceles Triangle TheoremThe converse of a conditional statement is made by swapping the hypothesis (if …) with the conclusion (then …). You may need to tinker with it to ensure it makes sense. So here once again is the Isosceles Triangle Theorem: If two sides of a triangle are congruent, then angles opposite those sides are congruent. To make its converse, we could exactly swap the parts, getting a bit of a mish-mash: If angles opposite those sides are congruent, then two sides of a triangle are congruent. That is awkward, so tidy up the wording:
The Converse of the Isosceles Triangle Theorem states: If two angles of a triangle are congruent, then sides opposite those angles are congruent. Now it makes sense, but is it true? Not every converse statement of a conditional statement is true. If the original conditional statement is false, then the converse will also be false. If the premise is true, then the converse could be true or false: If I see a bear, then I will lie down and remain still. If I lie down and remain still, then I will see a bear. For that converse statement to be true, sleeping in your bed would become a bizarre experience. Or this one: If I have honey, then I will attract bears. If I attract bears, then I will have honey. Unless the bears bring honeypots to share with you, the converse is unlikely ever to happen. And bears are famously selfish. Proving the Converse StatementTo prove the converse, let's construct another isosceles triangle, △BER. Given that ∠BER ≅ ∠BRE, we must prove that BE ≅ BR. Add the angle bisector from ∠EBR down to base ER. Where the angle bisector intersects base ER, label it Point A. Now we have two small, right triangles where once we had one big, isosceles triangle: △BEA and △BAR. Since line segment BA is an angle bisector, this makes ∠EBA ≅ ∠RBA. Since line segment BA is used in both smaller right triangles, it is congruent to itself. What do we have?
Let's see … that's an angle, another angle, and a side. That would be the Angle Angle Side Theorem, AAS:
The AAS Theorem states: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. With the triangles themselves proved congruent, their corresponding parts are congruent (CPCTC), which makes BE ≅ BR. The converse of the Isosceles Triangle Theorem is true! Lesson SummaryBy working through these exercises, you now are able to recognize and draw an isosceles triangle, mathematically prove congruent isosceles triangles using the Isosceles Triangles Theorem, and mathematically prove the converse of the Isosceles Triangles Theorem. You also should now see the connection between the Isosceles Triangle Theorem to the Side Side Side Postulate and the Angle Angle Side Theorem. Next Lesson:Alternate Exterior Angles
Instructor: Malcolm M. |
What theorem states that if two angles of a triangle are congruent then the sides opposite them are also congruent?
Pos Terkait
Copyright © 2024 ketajaman Inc.
