Question

warm up 🔥 get locked in!
basketball sam, kendra, and tony are passing a basketball
if sam is looking at kendra, then he needs to turn 40° to pass to tony. if
tony is looking at sam, then he needs to turn 50° to pass to kendra. how
many degrees would kendra have to turn her head to look at tony if she
is looking at sam?
kendra would have to turn her head ______ degrees
to look from sam to tony. i know this because
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real - world connection 🎯 why it matters!
the supports on the john e cox bridge in lowell, ma form right
triangles. the two remote interior angles of each triangle are 45°
45° = 90°, which equals the exterior angle formed by the beam
extending outward. this shows the exterior angle theorem in
action — which engineers use to make sure beams fit perfectly
that the structure is stable

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). Let the angles at Sam, Tony, and Kendra be \(S\), \(T\), and \(K\) respectively.
We know \(S = 40^\circ\) (Sam turns \(40^\circ\) to pass to Tony) and \(T = 50^\circ\) (Tony turns \(50^\circ\) to pass to Kendra).

Step2: Calculate Kendra's angle

Using the triangle angle sum formula \(S + T + K = 180^\circ\), substitute \(S = 40^\circ\) and \(T = 50^\circ\):
\(40^\circ+ 50^\circ+ K = 180^\circ\)
\(90^\circ+ K = 180^\circ\)
Subtract \(90^\circ\) from both sides: \(K = 180^\circ - 90^\circ=90^\circ\)? Wait, no, wait. Wait, the angles at Sam: when Sam looks at Kendra, turning \(40^\circ\) to Tony means the angle at Sam between Kendra and Tony is \(40^\circ\). At Tony, looking at Sam, turning \(50^\circ\) to Kendra means the angle at Tony between Sam and Kendra is \(50^\circ\). So the triangle has angles: at Sam: \(40^\circ\), at Tony: \(50^\circ\), so at Kendra: \(180 - 40 - 50 = 90^\circ\)? Wait, no, maybe I misinterpret. Wait, the problem is about Kendra looking at Sam, then turning to Tony. So the angle at Kendra between Sam and Tony is what we need. So in triangle Sam - Kendra - Tony, the angles at Sam (between Kendra and Tony) is \(40^\circ\), at Tony (between Sam and Kendra) is \(50^\circ\), so angle at Kendra is \(180 - 40 - 50 = 90^\circ\)? Wait, no, let's re-express. Let’s denote:

• When Sam is looking at Kendra, to pass to Tony, he turns \(40^\circ\): so \(\angle SKT = 40^\circ\) (angle at Sam between Kendra and Tony).

• When Tony is looking at Sam, to pass to Kendra, he turns \(50^\circ\): so \(\angle TSK = 50^\circ\) (angle at Tony between Sam and Kendra).

Then in \(\triangle SKT\), the sum of angles is \(180^\circ\). So angle at Kendra (\(\angle SKT\)? Wait, no, vertices are S (Sam), K (Kendra), T (Tony). So angles:

• \(\angle at S\): between K and T: \(40^\circ\)

• \(\angle at T\): between S and K: \(50^\circ\)

• \(\angle at K\): between S and T:?

So \(\angle S + \angle T + \angle K = 180^\circ\)

\(40^\circ + 50^\circ + \angle K = 180^\circ\)

\(\angle K = 180 - 40 - 50 = 90^\circ\). Wait, but that seems high. Wait, maybe the angles are: the angle Sam needs to turn from Kendra to Tony is \(40^\circ\), so that's the angle at Sam between K and T: \(\angle KST = 40^\circ\). The angle Tony needs to turn from Sam to Kendra is \(50^\circ\), so that's the angle at Tony between S and K: \(\angle STK = 50^\circ\). Then angle at K (\(\angle SKT\)) is \(180 - 40 - 50 = 90^\circ\). So Kendra, looking at Sam, needs to turn \(90^\circ\) to look at Tony. Wait, but maybe I made a mistake. Wait, let's check again. The sum of angles in a triangle is \(180^\circ\). So if two angles are \(40^\circ\) and \(50^\circ\), the third is \(90^\circ\). So Kendra's turn is \(90^\circ\).

Wait, no, maybe the angles are: Sam's angle: when he looks at Kendra, then turns \(40^\circ\) to Tony: so the angle between Kendra (his line of sight) and Tony (his line of sight) is \(40^\circ\), so that's the angle at Sam: \(\angle KST = 40^\circ\). Tony's angle: when he looks at Sam, turns \(50^\circ\) to Kendra: so the angle between Sam (his line of sight) and Kendra (his line of sight) is \(50^\circ\), so \(\angle STK = 50^\circ\). Then in triangle KST, angles sum to \(180^\circ\), so \(\angle SKT = 180 - 40 - 50 = 90^\circ\). So Kendra, looking at Sam (line of sight SK), needs to turn \(90^\circ\) to look at Tony (line of sight TK). So the answer is \(90\).

Answer:

90