10th Grade

How Do You Prove Vertical Angles Are Congruent?

{
"voice_prompt": "",
"manuscript": {
"title": {
"text": "How Do You Prove Vertical Angles Are Congruent?",
"audio": "How do you prove vertical angles are congruent?"
},
"description": {
"text": "Vertical angles are congruent because they form equal pairs around a point where two lines intersect. You can prove this using straight angles and subtraction.",
"audio": "Vertical angles are congruent because they form equal pairs around a point where two lines intersect. You can prove this using straight angles and subtraction."
},
"scenes": [
{
"text": "When two lines cross, they form four angles. These angles all meet at the same point, called the vertex. The angles that are directly across from each other, those that do not share a side, are called vertical angles. Your goal is to prove that vertical angles always have the same measure, or in math terms, that they are congruent.",
"latex": "\\text{Vertical angles: } \\angle 1 \\text{ and } \\angle 3, \\quad \\angle 2 \\text{ and } \\angle 4"
},
{
"text": "Focus on one pair of vertical angles: angle 1 and angle 3. Notice that angle 1 and angle 2 are next to each other and lie along a straight line. This kind of pair is called a linear pair. Because they form a straight angle, their measures must add up to 180 degrees. This is always true for angles that sit along a straight line—they form a half-turn around the point.",
"latex": "\\angle 1 + \\angle 2 = 180^\\circ"
},
{
"text": "Now look at angle 2 and angle 3. These two also form a linear pair because they sit next to each other along the same straight line. That means angle 2 plus angle 3 also equals 180 degrees.",
"latex": "\\angle 2 + \\angle 3 = 180^\\circ"
},
{
"text": "Since both expressions equal 180 degrees, you can set the two sides equal to each other: angle 1 plus angle 2 equals angle 2 plus angle 3.",
"latex": "\\angle 1 + \\angle 2 = \\angle 2 + \\angle 3"
},
{
"text": "Now subtract angle 2 from both sides of the equation. This step works because you're removing the same amount from both sides. You are left with angle 1 equals angle 3.",
"latex": "\\angle 1 = \\angle 3"
},
{
"text": "This proves that vertical angles—like angle 1 and angle 3—are congruent. That means they are always equal in size, no matter how the lines are angled.",
"latex": "\\text{Therefore, } \\angle 1 \\cong \\angle 3"
},
{
"text": "You can apply the exact same reasoning to the other pair of vertical angles: angle 2 and angle 4. Angle 1 and angle 2 form a linear pair, and so do angle 1 and angle 4.",
"latex": "\\angle 1 + \\angle 2 = 180^\\circ, \\quad \\angle 1 + \\angle 4 = 180^\\circ"
},
{
"text": "Since both sums equal 180 degrees, angle 2 must equal angle 4. That means angle 2 and angle 4 are also congruent.",
"latex": "\\angle 2 = \\angle 4 \\quad \\Rightarrow \\quad \\angle 2 \\cong \\angle 4"
},
{
"text": "To picture this in the real world, think of a pair of open scissors. The blades cross at a point, forming two angles on opposite sides. These angles are vertical angles—and they are equal, just like in your proof.",
"latex": "\\text{Scissors example: } \\angle A = \\angle B \\text{ (vertical angles)}"
}
],
"outro": {
"text": "Vertical angles are congruent because they form equal pairs around a point where two lines intersect. You can prove this using straight angles and subtraction.",
"audio": "Vertical angles are congruent because they form equal pairs around a point where two lines intersect. You can prove this using straight angles and subtraction."
}
}
}