Let's work through both problems in the image step by step.

Problem 15:

The quadrilateral shown is a trapezoid, and we are asked to find the value of $x$ and the measures of all labeled angles.

Given:

• One angle is $90^\circ$ (right angle).
• One angle is labeled as $x^\circ$.
• The other angle is labeled as $(x - 50)^\circ$.

In any quadrilateral, the sum of the interior angles is $360^\circ$.

Let's call the angles as follows:

• The right angle: $90^\circ$.
• The opposite angle: $x^\circ$.
• The remaining two angles: $(x - 50)^\circ$ and another $90^\circ$.

So, we can set up the equation:

$90^\circ + 90^\circ + x^\circ + (x - 50)^\circ = 360^\circ$

Simplifying the equation:

$180^\circ + x + x - 50 = 360^\circ$ $2x - 50 = 180^\circ$ $2x = 230^\circ$ $x = 115^\circ$

Thus, $x = 115^\circ$.

Angle measures:

• $x = 115^\circ$
• $(x - 50) = 115^\circ - 50^\circ = 65^\circ$
• The right angles are $90^\circ$ each.

Problem 16:

This figure is an isosceles triangle, and we are asked to find $x$ and the labeled angles.

Given:

• One angle is $(3x - 10)^\circ$.
• Another angle is $(x + 40)^\circ$.
• The third angle (the top vertex) appears unlabeled, but since the sum of angles in a triangle is $180^\circ$, we can use that information.

The sum of the interior angles of a triangle is always $180^\circ$. So, we can set up the equation:

$(3x - 10)^\circ + (x + 40)^\circ + (x + 40)^\circ = 180^\circ$

Simplifying the equation:

$3x - 10 + x + 40 + x + 40 = 180^\circ$ $5x + 70 = 180^\circ$ $5x = 110^\circ$ $x = 22^\circ$

Angle measures:

• $x = 22^\circ$
• $(x + 40) = 22^\circ + 40^\circ = 62^\circ$
• $(3x - 10) = 3(22^\circ) - 10^\circ = 66^\circ - 10^\circ = 56^\circ$

Summary:

1. For problem 15: $x = 115^\circ$, angles are $115^\circ$, $65^\circ$, and two $90^\circ$ angles.
2. For problem 16: $x = 22^\circ$, angles are $62^\circ$, $62^\circ$, and $56^\circ$.

Would you like any further clarifications on these solutions?

Further Related Questions:

1. How do you find the sum of the interior angles of any polygon?
2. What are the properties of an isosceles triangle that can help solve angle problems?
3. How can we verify angle relationships in a quadrilateral like a trapezoid?
4. How do angle sum properties change for concave vs. convex polygons?
5. What are the steps to solve problems involving supplementary and complementary angles?

Tip:

Remember that the sum of interior angles of any polygon can be found using the formula $(n-2) \times 180^\circ$, where $n$ is the number of sides.