Let's solve each question step-by-step and provide explanations using the relevant theorems.

Problem 3

Triangle Properties and Base Angles Theorem

In this triangle, we have two base angles, one labeled as $35^\circ$ and the other angle expression $3x + 20$. Since the triangle appears to be isosceles (indicated by the marks on the sides), the base angles are congruent.

1. Step 1: Set up the equation $3x + 20 = 35$

2. Step 2: Solve for $x$ $3x = 35 - 20$ $3x = 15$ $x = 5$

Explanation: Using the Base Angles Theorem, we know that in an isosceles triangle, the base angles are equal. Therefore, we equate $3x + 20$ to $35$ and solve for $x$.

Problem 4

Using the Triangle Sum Theorem

In this triangle, two angles are given: $50^\circ$ and the other angle marked as $x$. Since it is also isosceles (shown by the side marks), the two angles opposite the equal sides are equal.

1. Step 1: Set up the equation using Triangle Sum Theorem Since the sum of angles in a triangle is $180^\circ$, and the triangle has two equal angles $x$: $50 + x + x = 180$ $50 + 2x = 180$

2. Step 2: Solve for $x$ $2x = 180 - 50$ $2x = 130$ $x = 65$

Explanation: The Triangle Sum Theorem states that the sum of the interior angles of a triangle is $180^\circ$. Given that this triangle is isosceles, the two unknown angles are equal, allowing us to set up and solve the equation above.

Problem 5

Using the Triangle Sum Theorem and Base Angles Theorem

In this triangle, one angle is $40^\circ$, the other angle is $2x$, and the third angle is $y - 12$.

1. Step 1: Set up the equation The sum of the angles in a triangle is $180^\circ$. $40 + 2x + (y - 12) = 180$

2. Simplify and solve for $y$ in terms of $x$: $2x + y - 12 + 40 = 180$ $2x + y + 28 = 180$ $2x + y = 152$

However, without further information, we need additional conditions to solve this fully.

If you need further explanation or additional solutions, let me know.

Related Questions for Deeper Understanding:

1. What other types of triangles can use the Base Angles Theorem?
2. How does the Exterior Angle Theorem apply to these problems?
3. How would you solve similar triangles with exterior angles?
4. What changes if the triangle is not isosceles?
5. How do we solve for unknown angles in scalene triangles?

Tip: Always verify if the triangle is isosceles or equilateral before applying specific theorems related to equal angles.