Given: R S ≅R T, ∠S ≅∠T Prove: R S T ≅R T S

Name: Problem 37, Chapter 4: Geometry
Uploaded: 2020-03-28T05:18:46-08:00
Duration: 5 min 43 s
Channel: Jason Taylor-Pestell
Description: Given: R S ≅R T, ∠S ≅∠T Prove: R S T ≅R T S

Key Concepts

Reflexive Property

In geometry, the reflexive property refers to the idea that any geometric figure or segment is congruent to itself. This property is used in proofs to justify the equality of a shared part between two figures, thereby helping establish the overall congruence between the figures.

Isosceles Triangle Properties

An isosceles triangle is defined by having two sides of equal length, which in turn forces the base angles (the angles opposite these sides) to be congruent. This symmetry property is widely used in geometric arguments, as it often helps in establishing further equalities within the shape or its subdivided parts.

Congruence of Triangles

This concept involves proving that two triangles are identical in shape and size. It is established by demonstrating that certain corresponding parts, such as sides and angles, are equal in measure. Proving triangle congruence is a fundamental method in geometry for showing that two figures can be perfectly superimposed on each other through rigid motions.

Side-Angle-Side (SAS) Congruence Criterion

The SAS criterion states that if two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the triangles are congruent. This criterion is a powerful tool in geometric proofs because it directly links both linear and angular measurements within the triangles.

Transcript

00:01 In this problem, we are given that the length r .s is congruent to the length rt.

00:10 We are also given that angle s is congruent to angle t.

00:15 Based off of the diagram, then we could classify this as an isosceles triangle.

00:21 And we'll learn a lot more later down the road about isosceles triangles and their properties.

00:26 This is a rather interesting problem because we are proving that if you name the triangle, a specific way, then it's congruent to a triangle named a different way, which is technically then a different triangle.

00:44 Triangle rst is a different triangle than triangle rts, especially when we're talking about congruence because order matters a lot.

00:54 And so what we're going to do is i'm going to match up all of the corresponding parts.

00:59 I've already started with the first triangle.

01:02 So i have angle s.

01:04 Sorry, angle r, angle s, angle t, side length rs, then s t, then t r.

01:13 And so i'm going to set those to be congruent with their corresponding parts based out of this statement that we want to prove, and then we'll give justification for each of those different statements.

01:28 Since this is kind of a strange problem, i want to make sure i'm keeping it in order.

01:32 So angle r has to correspond with angle r, which is really nice.

01:38 We'll prove that in a moment why that's true.

01:42 The next is the middle letters for each.

01:44 So that's s and t.

01:47 So s has to correspond to t.

01:50 T has to correspond to s because those are the last two letters in the congruent statement.

01:59 So let's pause there and prove each of these statements.

02:02 So the first statement is saying that angle r is congruent to angle r.

02:08 And that's because it's in both of the triangles and it is the corresponding angle for each one.

02:16 Well, it is in both in itself and therefore that's reflexive property.

02:20 So this is congruent to itself by reflexive property of congruence.