Additive Angles

Wow! Check out Nico and Nia's new hang gliders! Each one of them thinks they can make it go higher so they decide to have a FRIENDLY competition! As their gliders lift off the ground, we can see that their flight patterns have created several angles. These angles are ADDITIVE, which means we can use addition and subtraction to solve for the unknown angle measurements. "Additive Angles" Nico and Nia's flights have created ADJACENT ANGLES. Let's look at the angles and see the line and the vertex they share. Adjacent angles share the SAME VERTEX and ONE SIDE. Adjacent angles have smaller angles and also a LARGER one that is created when they are COMBINED. Let’s measure the adjacent angles made by Nico and Nia's flights. Nia’s flight made the first angle. We know that it measures twenty degrees... ...And Nico flew twenty-five degrees HIGHER. How can we solve to find the TOTAL angle of Nico’s flight? We know the parts, so we can ADD to find the whole angle. What is the sum of twenty and twenty-five? Zero plus five is five, and two plus two make four. Forty-five! Nico’s flight was at a forty-five-degree angle! NICO WON! "No, no, no!" "It's the BEST OUT OF THREE That's how we decide the winner!" For the NEXT round, they decide to take off from a hill to give themselves more height. What observations can you make about the angles made by where they took off and the directions they went in? We can see the widest angle is made from the tree to the bottom of the hill, and Nia and Nico have now created a total of THREE angles inside. Nico flew at a sixty-six-degree angle off the ground. Nia's take-off was seventeen degrees higher than Nico's, (...) AND her flight path was twenty-six degrees from the tree. What do we need to do to find the angle measurement made by the tree to the ground? We need to ADD the three angles to find the sum. What is the total of sixty-six, plus seventeen, plus twenty-six? In the ones, six plus seven is thirteen. What is six more? Nineteen. In the tens, add six plus one plus two to make nine, (...) and now add one more. The total angle is one hundred nine degrees. Since Nia flew higher in round two, it's time to get SERIOUS for the tiebreaker! BLAST OFF! Nico and Nia took off in opposite directions and have created new adjacent angles with the ground. What do you observe about these angles? Here we have given the total of the widest angle, and the measurement of Nia's angle. What do you think we need to do to find the MISSING angle, which is the DIFFERENCE between Nico and Nia’s flight? In this problem, we will need to SUBTRACT the KNOWN angle from the WHOLE to find the missing angle. This means whole minus part equals part. We set up the problem, one hundred forty MINUS thirty. Start subtracting with the ones. There are zero ones. Four minus three equals one. We still have one in the hundreds because we didn’t have any to take away. The unknown angle measures one hundred ten degrees. Remember, Adjacent angles are ADDITIVE, so we can use known angle measurements to solve for the unknown parts. In a group of adjacent angles that are joined together by one vertex, we can add the smaller angles to find the sum of the degrees of the widest angle. Or sometimes, we know the combined angle measurement and we need to find a missing part. We solve these by taking the angle degrees we do know and subtracting that number from the whole. [Nia's about to fall] "WOAH!WOAH!WOAH!" "NICO!"HELP!" "NIA!! HANG ON!" "I'M ON MY WAY!" “Thank you, Nico!” “DO OVER?”