Math and Logic

A Triangle's Shadow

We can think of an angle as the measure of a turning motion or rotation.

When multiple angles are in the same diagram, they can be related to one another in several ways. We can use these relationships as tools for solving angle hunt problems. Keep reading to see some of these tools used, or jump ahead to today's challenge if you're ready.

Here are some ways we can relate angles:

1. When angles combine to form a straight line, their measures add to 180.180^\circ.

2. When two lines intersect, opposite angles have equal measures.

3. The measures of a triangle's interior angles add to 180.180^\circ.

Let's see some ways we can use these relationships. We'll start by finding the measure of the missing angle below.

We see three angles that combine to form a straight line. By Fact 1,1, we know that their measures add to 180.180^\circ. Additionally, the black square indicates a right angle, which has a measure of 90.90^\circ. So, we know that 42 + ?+90=180. 42\ +\ ? + 90 = 180. Adding 4242 and 9090 gives us 132,132, so the missing angle has measure 132132^\circ less than 180.180^\circ. Subtracting 132132^\circ from 180180^\circ gives us 48.48^\circ.

The next problem is a bit more complex. How can we find the value of x?x?

There are two triangles in the figure, so maybe Fact 33 will be helpful. Applying this fact to the triangle on the left won't tell us much, but applying it to the triangle on the right will let us find the measure of its unmarked angle. The measures of all three angles of the triangle add to 180,180^\circ, with the two marked angles' measures adding to 60+80=140.60^\circ + 80^\circ = 140^\circ. This means the unmarked angle must have a measure of 180140=40.180^\circ - 140^\circ = 40^\circ.

Fact 22 tells us that the left triangle's unmarked angle also has a measure of 40.40^\circ. Now that we know this, Fact 33 should be more helpful. We can relate the measures of all three angles of the triangle by the sum x+x+40=180. x + x + 40 = 180. This means x+xx + x is equal to 18040=140,180 - 40 = 140, so x=140÷2=70.x = 140 \div 2 = 70.

See if you can use these angle relationships to tackle the angle hunt challenge below.

Today's Challenge

The two gray angles have the same measure. How are the angles x,y,x^\circ, y^\circ, and zz^\circ related?


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