Use the diagram (FIGURE CANT COPY) Given that R Q=T Q, m ∠Q S R=(9 a+48)^∘ and m ∠Q S T=(6 a+50)

step 1 For problem 17, we're given a figure. We're also given that RQ equals TQ, which means segment RQ

Use the diagram (FIGURE CANT COPY) Given that R Q=T Q, m ∠Q...

Key Concepts

Isosceles Triangle

An isosceles triangle is a triangle that has at least two congruent sides. One of the key properties of isosceles triangles is that the angles opposite the equal sides (called base angles) are congruent. This property is essential when determining unknown angle measures, as it allows the formation of equations based on the equality of angles.

Angle Relationships

Angle relationships involve the various ways angles interact with one another in geometric figures, such as adjacent angles, vertical angles, and complementary or supplementary angles. Understanding these relationships is crucial when angles are defined by algebraic expressions, as it allows one to set up equations based on angle sums or equalities imposed by the diagram's geometry.

Algebraic Methods in Geometry

Algebraic methods in geometry involve representing geometric measures, such as angles or sides, with algebraic expressions and using equations to solve for unknown values. This approach leverages the properties of geometric figures (like isosceles triangles or angle sum properties) and allows one to solve for variables that determine specific measurements in the figure.

Transcript

00:01 For problem 17, we're given a figure.

00:03 We're also given that rq equals tq, which means segment rq is equal to segment tq.

00:10 So we can show that by using our tick marks.

00:14 Angle qsr, which is this angle here, is 9a plus 48 degrees.

00:25 And angle qst, which is this angle here, is 6a plus 50 degrees.

00:32 We can't assume that these angles are congruent, so we use different labels for them.

00:39 Whoops, i put 58.

00:41 There we go.

00:43 They want us to find the measure of angle qst, which is this angle here.

00:48 So that's good news.

00:49 All we need to do is figure out what a is, so we can plug that in to figure out the whole angle qst.

00:56 Just gotta plug it into the expression.

01:00 So we need to figure out what we can do, especially with this 9a plus 48.

01:07 We want these angles to be equal, but we have to prove it.

01:10 And when we're proving things, we definitely like to use theorems.

01:14 So the theorem we want to use for this problem is the converse of the angle bisector theorem.

01:20 The converse of the angle bisector theorem says that if a point is in the interior of the angle, so q is on the interior of the angle, the whole angle is rst that we're looking at.

01:34 So q is on the inside.

01:37 If it's equidistant from the sides of the angle.

01:40 So the sides are rs and st.

01:43 Q has to be the same length away from rs as it is from st, which in this case it is because from q to r is the same length as from q to t.

01:56 We know that because they gave it to us in the beginning.

01:59 So q is on the interior.

02:01 It's equidistant from the sides, so that means that it's on an angle bisector.

02:05 So that means that sq, segment sq, is an angle bisector.

02:10 Well, that is awesome news because we know an angle bisector cuts the angle into two equal parts, which means that angle qsr is congruent to angle qst.

02:28 So now we only do need the same amount of tick marks.

02:32 I'm just going to make a second one here...

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