Question: When did the party with a majority in the body the President calls on for support of his appointments to the USAF gain control of the House?

Knowledge 1: United States Air Force
The Department of the Air Force is one of three military departments within the Department of Defense, and is managed by the civilian Secretary of the Air Force, under the authority, direction, and control of the Secretary of Defense. The senior officials in the Office of the Secretary are the Under Secretary of the Air Force, four Assistant Secretaries of the Air Force and the General Counsel, all of whom are appointed by the President with the advice and consent of the Senate. The senior uniformed leadership in the Air Staff is made up of the Chief of Staff of the Air Force and the Vice Chief of Staff of the Air Force. [SEP] Knowledge 2: Dwight D. Eisenhower
The Democrats gained a majority in both houses in the 1954 election. Eisenhower had to work with the Democratic Majority Leader Lyndon B. Johnson (later U.S. president) in the Senate and Speaker Sam Rayburn in the House, both from Texas. Joe Martin, the Republican Speaker from 1947 to 1949 and again from 1953 to 1955, wrote that Eisenhower "never surrounded himself with assistants who could solve political problems with professional skill. There were exceptions, Leonard W. Hall, for example, who as chairman of the Republican National Committee tried to open the administration's eyes to the political facts of life, with occasional success. However, these exceptions were not enough to right the balance." [SEP] Knowledge 3: Party leaders of the United States Senate
The Senate is currently composed of 51 Republicans, 47 Democrats, and 2 independents, both of whom caucus with the Democrats.
Knowledge 1: United States Air Force
The Department of the Air Force is one of three military departments within the Department of Defense, and is managed by the civilian Secretary of the Air Force, under the authority, direction, and control of the Secretary of Defense. The senior officials in the Office of the Secretary are the Under Secretary of the Air Force, four Assistant Secretaries of the Air Force and the General Counsel, all of whom are appointed by the President with the advice and consent of the Senate. The senior uniformed leadership in the Air Staff is made up of the Chief of Staff of the Air Force and the Vice Chief of Staff of the Air Force. [SEP] Knowledge 2: Dwight D. Eisenhower
The Democrats gained a majority in both houses in the 1954 election. Eisenhower had to work with the Democratic Majority Leader Lyndon B. Johnson (later U.S. president) in the Senate and Speaker Sam Rayburn in the House, both from Texas. Joe Martin, the Republican Speaker from 1947 to 1949 and again from 1953 to 1955, wrote that Eisenhower "never surrounded himself with assistants who could solve political problems with professional skill. There were exceptions, Leonard W. Hall, for example, who as chairman of the Republican National Committee tried to open the administration's eyes to the political facts of life, with occasional success. However, these exceptions were not enough to right the balance." [SEP] Knowledge 3: Party leaders of the United States House of Representatives
Starting with Republican Nicholas Longworth in 1925, and continued through the Democrats' control of the House from 1931 to 1995, save for Republican majorities in 1947–49 and 1953–55, all majority leaders have directly ascended to the Speakership brought upon by the retirement of the incumbent. The only exceptions during this period were Charles A. Halleck who became Republican House leader and Minority Leader from 1959 to 1965, Hale Boggs who died in a plane crash, and Dick Gephardt who became the Democrats' House leader but as Minority Leader since his party lost control in the 1994 midterm elections. Since 1995, the only Majority Leader to become Speaker is John Boehner, though indirectly as his party lost control in the 2006 midterms elections. He subsequently served as Republican House leader and Minority Leader from 2007 to 2011 and then was elected Speaker when the House reconvened in 2011. In 1998, with Speaker Newt Gingrich announcing his resignation, both Majority Leader Dick Armey and Majority Whip Tom DeLay did not contest the Speakership which eventually went to Chief Deputy Whip Dennis Hastert.
Knowledge 1: United States Air Force
The Department of the Air Force is one of three military departments within the Department of Defense, and is managed by the civilian Secretary of the Air Force, under the authority, direction, and control of the Secretary of Defense. The senior officials in the Office of the Secretary are the Under Secretary of the Air Force, four Assistant Secretaries of the Air Force and the General Counsel, all of whom are appointed by the President with the advice and consent of the Senate. The senior uniformed leadership in the Air Staff is made up of the Chief of Staff of the Air Force and the Vice Chief of Staff of the Air Force. [SEP] Knowledge 2: Dwight D. Eisenhower
The Democrats gained a majority in both houses in the 1954 election. Eisenhower had to work with the Democratic Majority Leader Lyndon B. Johnson (later U.S. president) in the Senate and Speaker Sam Rayburn in the House, both from Texas. Joe Martin, the Republican Speaker from 1947 to 1949 and again from 1953 to 1955, wrote that Eisenhower "never surrounded himself with assistants who could solve political problems with professional skill. There were exceptions, Leonard W. Hall, for example, who as chairman of the Republican National Committee tried to open the administration's eyes to the political facts of life, with occasional success. However, these exceptions were not enough to right the balance." [SEP] Knowledge 3: Party leaders of the United States House of Representatives
The current Minority Leader Nancy Pelosi, of the United States House of Representatives serves as floor leader of the opposition party, and is the counterpart to the Majority Leader. Unlike the Majority Leader, the Minority Leader is on the ballot for Speaker of the House during the convening of the Congress. If the Minority Leader's party takes control of the House, and the party officers are all re-elected to their seats, the Minority Leader is usually the party's top choice for Speaker for the next Congress, while the Minority Whip is typically in line to become Majority Leader. The Minority Leader usually meets with the Majority Leader and the Speaker to discuss agreements on controversial issues.

Answer: the 2010 election